Theorem frege64c 44653

Description: Lemma for frege65c 44654. 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 44651	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥]𝜒))
3		frege18 44544	. 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 1568   ∈ wcel 2143  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-frege1 44516  ax-frege2 44517  ax-frege8 44535  ax-frege58b 44627

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-sbc 3745

This theorem is referenced by:  frege65c  44654