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Theorem frege72 40271
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 40257 . . 3 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → [𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴))
3 sbcim1 3823 . . . 4 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → ([𝑌 / 𝑧]𝑋𝑅𝑧[𝑌 / 𝑧]𝑧𝐴))
4 sbcbr2g 5115 . . . . . 6 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌 / 𝑧𝑧))
5 csbvarg 4381 . . . . . . 7 (𝑌𝑉𝑌 / 𝑧𝑧 = 𝑌)
65breq2d 5069 . . . . . 6 (𝑌𝑉 → (𝑋𝑅𝑌 / 𝑧𝑧𝑋𝑅𝑌))
74, 6bitrd 281 . . . . 5 (𝑌𝑉 → ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌))
81, 7ax-mp 5 . . . 4 ([𝑌 / 𝑧]𝑋𝑅𝑧𝑋𝑅𝑌)
9 sbcel1v 3837 . . . 4 ([𝑌 / 𝑧]𝑧𝐴𝑌𝐴)
103, 8, 93imtr3g 297 . . 3 ([𝑌 / 𝑧](𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
112, 10syl 17 . 2 (∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴))
12 frege72.x . . 3 𝑋𝑈
1312frege71 40270 . 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 208  wal 1528  wcel 2107  [wsbc 3770  csb 3881   class class class wbr 5057   hereditary whe 40108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-frege1 40126  ax-frege2 40127  ax-frege8 40145  ax-frege52a 40193  ax-frege58b 40237
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1057  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-he 40109
This theorem is referenced by:  frege73  40272  frege74  40273
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