Theorem frege56c 40620

Description: Lemma for frege57c 40621. 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 40616	. . . 4 ⊢ [𝐵 / 𝑥]𝑥 = 𝐵
3		frege53c 40615	. . . 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 40617	. . 3 ⊢ ((𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴 → 𝐴 = 𝐵))
6	4, 5	ax-mp 5	. 2 ⊢ (𝐵 = 𝐴 → 𝐴 = 𝐵)
7		frege9 40513	. 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 1538   ∈ wcel 2111  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-frege1 40491  ax-frege2 40492  ax-frege8 40510  ax-frege52c 40589

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721  df-sn 4526

This theorem is referenced by:  frege57c  40621