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Theorem frege73 43839
Description: Lemma for frege87 43853. 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 43838 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
4 ax-frege2 43694 . 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 2103   class class class wbr 5169   hereditary whe 43675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-frege1 43693  ax-frege2 43694  ax-frege8 43712  ax-frege52a 43760  ax-frege58b 43804
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-cnv 5707  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-he 43676
This theorem is referenced by:  frege87  43853
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