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Theorem frege72 42281
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 42267 . . 3 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴))
3 sbcim1 3800 . . . 4 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧[𝑌 / 𝑧]𝑧𝐴))
4 sbcbr2g 5168 . . . . . 6 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌 / 𝑧𝑧))
5 csbvarg 4396 . . . . . . 7 (𝑌𝑉𝑌 / 𝑧𝑧 = 𝑌)
65breq2d 5122 . . . . . 6 (𝑌𝑉 → (𝑋𝑅𝑌 / 𝑧𝑧𝑋𝑅𝑌))
74, 6bitrd 279 . . . . 5 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌))
81, 7ax-mp 5 . . . 4 ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌)
9 sbcel1v 3815 . . . 4 ([𝑌 / 𝑧]𝑧𝐴𝑌𝐴)
103, 8, 93imtr3g 295 . . 3 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
112, 10syl 17 . 2 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
12 frege72.x . . 3 𝑋𝑈
1312frege71 42280 . 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 205  wal 1540  wcel 2107  [wsbc 3744  csb 3860   class class class wbr 5110   hereditary whe 42118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-frege1 42136  ax-frege2 42137  ax-frege8 42155  ax-frege52a 42203  ax-frege58b 42247
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-he 42119
This theorem is referenced by:  frege73  42282  frege74  42283
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