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Theorem frege63c 37043
Description: Analogue of frege63b 37025. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege63c ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))

Proof of Theorem frege63c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege62c 37042 . 2 ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))
3 frege24 36932 . 2 (([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))))
42, 3ax-mp 5 1 ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wcel 1976  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-13 2232  ax-ext 2589  ax-frege1 36907  ax-frege2 36908  ax-frege8 36926  ax-frege58b 37018
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-v 3174  df-sbc 3402
This theorem is referenced by:  frege91  37071
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