Theorem frege66c 44771

Description: Swap antecedents of frege65c 44770. 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 44770	. 2 ⊢ (∀𝑥(𝜒 → 𝜑) → (∀𝑥(𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓)))
3		ax-frege8 44651	. 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)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   ∈ wcel 2145  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-frege1 44632  ax-frege2 44633  ax-frege8 44651  ax-frege58b 44743

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-sbc 3743

This theorem is used by: (None)