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Theorem frege72 39070
Description: If property 𝐴 is hereditary in the 𝑅-sequence, if 𝑥 has property 𝐴, and if 𝑦 is a result of an application of the procedure 𝑅 to 𝑥, then 𝑦 has property 𝐴. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege72.x 𝑋𝑈
frege72.y 𝑌𝑉
Assertion
Ref Expression
frege72 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))

Proof of Theorem frege72
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frege72.y . . . 4 𝑌𝑉
21frege58c 39056 . . 3 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴))
3 sbcim1 3710 . . . 4 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧[𝑌 / 𝑧]𝑧𝐴))
4 sbcbr2g 4932 . . . . . 6 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌 / 𝑧𝑧))
5 csbvarg 4228 . . . . . . 7 (𝑌𝑉𝑌 / 𝑧𝑧 = 𝑌)
65breq2d 4886 . . . . . 6 (𝑌𝑉 → (𝑋𝑅𝑌 / 𝑧𝑧𝑋𝑅𝑌))
74, 6bitrd 271 . . . . 5 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌))
81, 7ax-mp 5 . . . 4 ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌)
9 sbcel1v 3722 . . . 4 ([𝑌 / 𝑧]𝑧𝐴𝑌𝐴)
103, 8, 93imtr3g 287 . . 3 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
112, 10syl 17 . 2 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
12 frege72.x . . 3 𝑋𝑈
1312frege71 39069 . 2 ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
1411, 13ax-mp 5 1 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1656  wcel 2166  [wsbc 3663  csb 3758   class class class wbr 4874   hereditary whe 38907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-frege1 38925  ax-frege2 38926  ax-frege8 38944  ax-frege52a 38992  ax-frege58b 39036
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-xp 5349  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-he 38908
This theorem is referenced by:  frege73  39071  frege74  39072
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