Theorem frege63c 40278

Description: Analogue of frege63b 40260. 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 40277	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒))
3		frege24 40167	. 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 1534   ∈ wcel 2113  [wsbc 3775

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-12 2176  ax-ext 2796  ax-frege1 40142  ax-frege2 40143  ax-frege8 40161  ax-frege58b 40253

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3499  df-sbc 3776

This theorem is referenced by:  frege91  40306