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Theorem frege73 39069
Description: Lemma for frege87 39083. 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 39068 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
4 ax-frege2 38924 . 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 2164   class class class wbr 4875   hereditary whe 38905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege52a 38990  ax-frege58b 39034
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-he 38906
This theorem is referenced by:  frege87  39083
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