Theorem frege61c 37238

Description: Lemma for frege65c 37242. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege61c	⊢ (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))

Proof of Theorem frege61c

Step	Hyp	Ref	Expression
1		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
2	1	frege58c 37235	. 2 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)
3		frege9 37126	. 2 ⊢ ((∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) → (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))
4	2, 3	ax-mp 5	1 ⊢ (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1473   ∈ wcel 1977  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege58b 37215

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by:  frege65c  37242