Theorem frege57c 44704

Description: Swap order of implication in ax-frege52c 44672. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege57c.a	⊢ 𝐴 ∈ 𝐶

Assertion

Ref	Expression
frege57c	⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem frege57c

Step	Hyp	Ref	Expression
1		ax-frege52c 44672	. 2 ⊢ (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
2		frege57c.a	. . 3 ⊢ 𝐴 ∈ 𝐶
3	2	frege56c 44703	. 2 ⊢ ((𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)) → (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑)))
4	1, 3	ax-mp 5	1 ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  [wsbc 3746

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 2148  ax-9 2156  ax-ext 2737  ax-frege1 44574  ax-frege2 44575  ax-frege8 44593  ax-frege52c 44672

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-sbc 3747  df-sn 4592

This theorem is used by: (None)