Theorem frege66c 43390

Description: Swap antecedents of frege65c 43389. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege66c	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓)))

Proof of Theorem frege66c

Step	Hyp	Ref	Expression
1		frege59c.a	. . 3 ⊢ 𝐴 ∈ 𝐵
2	1	frege65c 43389	. 2 ⊢ (∀𝑥(𝜒 → 𝜑) → (∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓)))
3		ax-frege8 43270	. 2 ⊢ ((∀𝑥(𝜒 → 𝜑) → (∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓))) → (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓))))
4	2, 3	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4  ∀wal 1531   ∈ wcel 2098  [wsbc 3778

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 2699  ax-frege1 43251  ax-frege2 43252  ax-frege8 43270  ax-frege58b 43362

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-sbc 3779

This theorem is referenced by: (None)