Theorem frege63c 41211

Description: Analogue of frege63b 41193. 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 41210	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒))
3		frege24 41100	. 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 1541   ∈ wcel 2110  [wsbc 3694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-frege1 41075  ax-frege2 41076  ax-frege8 41094  ax-frege58b 41186

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-sbc 3695

This theorem is referenced by:  frege91  41239