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Theorem frege71 43947
Description: Lemma for frege72 43948. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege71.x 𝑋𝑉
Assertion
Ref Expression
frege71 ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋
Allowed substitution hints:   𝑉(𝑧)   𝑌(𝑧)

Proof of Theorem frege71
StepHypRef Expression
1 frege71.x . . 3 𝑋𝑉
21frege70 43946 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑧(𝑋𝑅𝑧𝑧𝐴)))
3 frege19 43837 . 2 ((𝑅 hereditary 𝐴 → (𝑋𝐴 → ∀𝑧(𝑋𝑅𝑧𝑧𝐴))) → ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))))
42, 3ax-mp 5 1 ((∀𝑧(𝑋𝑅𝑧𝑧𝐴) → (𝑋𝑅𝑌𝑌𝐴)) → (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wcel 2108   class class class wbr 5143   hereditary whe 43785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-frege1 43803  ax-frege2 43804  ax-frege8 43822  ax-frege52a 43870  ax-frege58b 43914
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-he 43786
This theorem is referenced by:  frege72  43948
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