Theorem frege57c 43882

Description: Swap order of implication in ax-frege52c 43850. 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 43850	. 2 ⊢ (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
2		frege57c.a	. . 3 ⊢ 𝐴 ∈ 𝐶
3	2	frege56c 43881	. 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))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-frege1 43752  ax-frege2 43753  ax-frege8 43771  ax-frege52c 43850

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-sbc 3805  df-sn 4649

This theorem is referenced by: (None)