Theorem frege64c 43251

Description: Lemma for frege65c 43252. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege64c	⊢ (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))

Proof of Theorem frege64c

Step	Hyp	Ref	Expression
1		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
2	1	frege62c 43249	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥]𝜒))
3		frege18 43142	. 2 ⊢ (([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥]𝜒)) → (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))))
4	2, 3	ax-mp 5	1 ⊢ (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))

Syntax hints:   → wi 4  ∀wal 1531   ∈ wcel 2098  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-frege1 43114  ax-frege2 43115  ax-frege8 43133  ax-frege58b 43225

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-sbc 3773

This theorem is referenced by:  frege65c  43252