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Theorem frege73 43369
Description: Lemma for frege87 43383. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege73.x 𝑋𝑈
frege73.y 𝑌𝑉
Assertion
Ref Expression
frege73 ((𝑅 hereditary 𝐴𝑋𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))

Proof of Theorem frege73
StepHypRef Expression
1 frege73.x . . 3 𝑋𝑈
2 frege73.y . . 3 𝑌𝑉
31, 2frege72 43368 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
4 ax-frege2 43224 . 2 ((𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))) → ((𝑅 hereditary 𝐴𝑋𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
53, 4ax-mp 5 1 ((𝑅 hereditary 𝐴𝑋𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5150   hereditary whe 43205
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-frege1 43223  ax-frege2 43224  ax-frege8 43242  ax-frege52a 43290  ax-frege58b 43334
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-he 43206
This theorem is referenced by:  frege87  43383
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