Theorem frege63c 44282

Description: Analogue of frege63b 44264. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege63c	⊢ ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒)))

Proof of Theorem frege63c

Step	Hyp	Ref	Expression
1		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
2	1	frege62c 44281	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒))
3		frege24 44171	. 2 ⊢ (([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒))))
4	2, 3	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒)))

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 2114  [wsbc 3742

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-frege1 44146  ax-frege2 44147  ax-frege8 44165  ax-frege58b 44257

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-sbc 3743

This theorem is referenced by:  frege91  44310