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Theorem frege77 37050
Description: If 𝑌 follows 𝑋 in the 𝑅-sequence, if property 𝐴 is hereditary in the 𝑅-sequence, and if every result of an application of the procedure 𝑅 to 𝑋 has the property 𝐴, then 𝑌 has property 𝐴. Proposition 77 of [Frege1879] p. 62. (Contributed by RP, 29-Jun-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege77.x 𝑋𝑈
frege77.y 𝑌𝑉
frege77.r 𝑅𝑊
frege77.a 𝐴𝐵
Assertion
Ref Expression
frege77 (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴)))
Distinct variable groups:   𝐴,𝑎   𝑅,𝑎   𝑋,𝑎
Allowed substitution hints:   𝐵(𝑎)   𝑈(𝑎)   𝑉(𝑎)   𝑊(𝑎)   𝑌(𝑎)

Proof of Theorem frege77
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 frege77.x . . 3 𝑋𝑈
2 frege77.y . . 3 𝑌𝑉
3 frege77.r . . 3 𝑅𝑊
41, 2, 3dffrege76 37049 . 2 (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)) ↔ 𝑋(t+‘𝑅)𝑌)
5 frege77.a . . . 4 𝐴𝐵
65frege68c 37041 . . 3 ((∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)) ↔ 𝑋(t+‘𝑅)𝑌) → (𝑋(t+‘𝑅)𝑌[𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓))))
7 sbcimg 3443 . . . . 5 (𝐴𝐵 → ([𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)) ↔ ([𝐴 / 𝑓]𝑅 hereditary 𝑓[𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓))))
85, 7ax-mp 5 . . . 4 ([𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)) ↔ ([𝐴 / 𝑓]𝑅 hereditary 𝑓[𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)))
9 sbcheg 36889 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑓]𝑅 hereditary 𝑓𝐴 / 𝑓𝑅 hereditary 𝐴 / 𝑓𝑓))
105, 9ax-mp 5 . . . . . 6 ([𝐴 / 𝑓]𝑅 hereditary 𝑓𝐴 / 𝑓𝑅 hereditary 𝐴 / 𝑓𝑓)
11 csbconstg 3511 . . . . . . . 8 (𝐴𝐵𝐴 / 𝑓𝑅 = 𝑅)
125, 11ax-mp 5 . . . . . . 7 𝐴 / 𝑓𝑅 = 𝑅
13 csbvarg 3954 . . . . . . . 8 (𝐴𝐵𝐴 / 𝑓𝑓 = 𝐴)
145, 13ax-mp 5 . . . . . . 7 𝐴 / 𝑓𝑓 = 𝐴
15 heeq12 36886 . . . . . . 7 ((𝐴 / 𝑓𝑅 = 𝑅𝐴 / 𝑓𝑓 = 𝐴) → (𝐴 / 𝑓𝑅 hereditary 𝐴 / 𝑓𝑓𝑅 hereditary 𝐴))
1612, 14, 15mp2an 703 . . . . . 6 (𝐴 / 𝑓𝑅 hereditary 𝐴 / 𝑓𝑓𝑅 hereditary 𝐴)
1710, 16bitri 262 . . . . 5 ([𝐴 / 𝑓]𝑅 hereditary 𝑓𝑅 hereditary 𝐴)
18 sbcimg 3443 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓) ↔ ([𝐴 / 𝑓]𝑎(𝑋𝑅𝑎𝑎𝑓) → [𝐴 / 𝑓]𝑌𝑓)))
195, 18ax-mp 5 . . . . . 6 ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓) ↔ ([𝐴 / 𝑓]𝑎(𝑋𝑅𝑎𝑎𝑓) → [𝐴 / 𝑓]𝑌𝑓))
20 sbcal 3451 . . . . . . . 8 ([𝐴 / 𝑓]𝑎(𝑋𝑅𝑎𝑎𝑓) ↔ ∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎𝑎𝑓))
21 sbcimg 3443 . . . . . . . . . . 11 (𝐴𝐵 → ([𝐴 / 𝑓](𝑋𝑅𝑎𝑎𝑓) ↔ ([𝐴 / 𝑓]𝑋𝑅𝑎[𝐴 / 𝑓]𝑎𝑓)))
225, 21ax-mp 5 . . . . . . . . . 10 ([𝐴 / 𝑓](𝑋𝑅𝑎𝑎𝑓) ↔ ([𝐴 / 𝑓]𝑋𝑅𝑎[𝐴 / 𝑓]𝑎𝑓))
23 sbcg 3469 . . . . . . . . . . . 12 (𝐴𝐵 → ([𝐴 / 𝑓]𝑋𝑅𝑎𝑋𝑅𝑎))
245, 23ax-mp 5 . . . . . . . . . . 11 ([𝐴 / 𝑓]𝑋𝑅𝑎𝑋𝑅𝑎)
25 sbcel2gv 3462 . . . . . . . . . . . 12 (𝐴𝐵 → ([𝐴 / 𝑓]𝑎𝑓𝑎𝐴))
265, 25ax-mp 5 . . . . . . . . . . 11 ([𝐴 / 𝑓]𝑎𝑓𝑎𝐴)
2724, 26imbi12i 338 . . . . . . . . . 10 (([𝐴 / 𝑓]𝑋𝑅𝑎[𝐴 / 𝑓]𝑎𝑓) ↔ (𝑋𝑅𝑎𝑎𝐴))
2822, 27bitri 262 . . . . . . . . 9 ([𝐴 / 𝑓](𝑋𝑅𝑎𝑎𝑓) ↔ (𝑋𝑅𝑎𝑎𝐴))
2928albii 1736 . . . . . . . 8 (∀𝑎[𝐴 / 𝑓](𝑋𝑅𝑎𝑎𝑓) ↔ ∀𝑎(𝑋𝑅𝑎𝑎𝐴))
3020, 29bitri 262 . . . . . . 7 ([𝐴 / 𝑓]𝑎(𝑋𝑅𝑎𝑎𝑓) ↔ ∀𝑎(𝑋𝑅𝑎𝑎𝐴))
31 sbcel2gv 3462 . . . . . . . 8 (𝐴𝐵 → ([𝐴 / 𝑓]𝑌𝑓𝑌𝐴))
325, 31ax-mp 5 . . . . . . 7 ([𝐴 / 𝑓]𝑌𝑓𝑌𝐴)
3330, 32imbi12i 338 . . . . . 6 (([𝐴 / 𝑓]𝑎(𝑋𝑅𝑎𝑎𝑓) → [𝐴 / 𝑓]𝑌𝑓) ↔ (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴))
3419, 33bitri 262 . . . . 5 ([𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓) ↔ (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴))
3517, 34imbi12i 338 . . . 4 (([𝐴 / 𝑓]𝑅 hereditary 𝑓[𝐴 / 𝑓](∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)) ↔ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴)))
368, 35bitri 262 . . 3 ([𝐴 / 𝑓](𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)) ↔ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴)))
376, 36syl6ib 239 . 2 ((∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝑋𝑅𝑎𝑎𝑓) → 𝑌𝑓)) ↔ 𝑋(t+‘𝑅)𝑌) → (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴))))
384, 37ax-mp 5 1 (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎𝑎𝐴) → 𝑌𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472   = wceq 1474  wcel 1976  [wsbc 3401  csb 3498   class class class wbr 4577  cfv 5790  t+ctcl 13518   hereditary whe 36882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-frege1 36900  ax-frege2 36901  ax-frege8 36919  ax-frege52a 36967  ax-frege58b 37011
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-seq 12619  df-trcl 13520  df-relexp 13555  df-he 36883
This theorem is referenced by:  frege78  37051  frege85  37058
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