Theorem frege61c 41905

Description: Lemma for frege65c 41909. 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 41902	. 2 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)
3		frege9 41793	. 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 1539   ∈ wcel 2106  [wsbc 3730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-frege1 41771  ax-frege2 41772  ax-frege8 41790  ax-frege58b 41882

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-sbc 3731

This theorem is referenced by:  frege65c  41909