Theorem frege56c 41416

Description: Lemma for frege57c 41417. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege56c.b	⊢ 𝐵 ∈ 𝐶

Assertion

Ref	Expression
frege56c	⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem frege56c

Step	Hyp	Ref	Expression
1		frege56c.b	. . . . 5 ⊢ 𝐵 ∈ 𝐶
2	1	frege54cor1c 41412	. . . 4 ⊢ [𝐵 / 𝑥]𝑥 = 𝐵
3		frege53c 41411	. . . 4 ⊢ ([𝐵 / 𝑥]𝑥 = 𝐵 → (𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵)
5		frege55lem1c 41413	. . 3 ⊢ ((𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴 → 𝐴 = 𝐵))
6	4, 5	ax-mp 5	. 2 ⊢ (𝐵 = 𝐴 → 𝐴 = 𝐵)
7		frege9 41309	. 2 ⊢ ((𝐵 = 𝐴 → 𝐴 = 𝐵) → ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))))
8	6, 7	ax-mp 5	1 ⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-frege1 41287  ax-frege2 41288  ax-frege8 41306  ax-frege52c 41385

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-sbc 3712  df-sn 4559

This theorem is referenced by:  frege57c  41417