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Theorem frege72 37073
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 37059 . . 3 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴))
3 sbcim1 3448 . . . 4 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧[𝑌 / 𝑧]𝑧𝐴))
4 sbcbr2g 4634 . . . . . 6 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌 / 𝑧𝑧))
5 csbvarg 3954 . . . . . . 7 (𝑌𝑉𝑌 / 𝑧𝑧 = 𝑌)
65breq2d 4589 . . . . . 6 (𝑌𝑉 → (𝑋𝑅𝑌 / 𝑧𝑧𝑋𝑅𝑌))
74, 6bitrd 266 . . . . 5 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌))
81, 7ax-mp 5 . . . 4 ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌)
9 sbcel1v 3461 . . . 4 ([𝑌 / 𝑧]𝑧𝐴𝑌𝐴)
103, 8, 93imtr3g 282 . . 3 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
112, 10syl 17 . 2 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
12 frege72.x . . 3 𝑋𝑈
1312frege71 37072 . 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 194  wal 1472  wcel 1976  [wsbc 3401  csb 3498   class class class wbr 4577   hereditary whe 36910
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-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-frege1 36928  ax-frege2 36929  ax-frege8 36947  ax-frege52a 36995  ax-frege58b 37039
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  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-ral 2900  df-rex 2901  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-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-he 36911
This theorem is referenced by:  frege73  37074  frege74  37075
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