Theorem frege56c 44500

Description: Lemma for frege57c 44501. 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 44496	. . . 4 ⊢ [𝐵 / 𝑥]𝑥 = 𝐵
3		frege53c 44495	. . . 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 44497	. . 3 ⊢ ((𝐵 = 𝐴 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴 → 𝐴 = 𝐵))
6	4, 5	ax-mp 5	. 2 ⊢ (𝐵 = 𝐴 → 𝐴 = 𝐵)
7		frege9 44393	. 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 1562   ∈ wcel 2144  [wsbc 3746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-frege1 44371  ax-frege2 44372  ax-frege8 44390  ax-frege52c 44469

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-sbc 3747  df-sn 4585

This theorem is referenced by:  frege57c  44501