Theorem frege63c 43915

Description: Analogue of frege63b 43897. 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 43914	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒))
3		frege24 43804	. 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 1538   ∈ wcel 2109  [wsbc 3753

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-frege1 43779  ax-frege2 43780  ax-frege8 43798  ax-frege58b 43890

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-sbc 3754

This theorem is referenced by:  frege91  43943