Theorem frege58c 39055

Description: Principle related to sp 2226. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege58c.a	⊢ 𝐴 ∈ 𝐵

Assertion

Ref	Expression
frege58c	⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)

Proof of Theorem frege58c

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege58c.a	. 2 ⊢ 𝐴 ∈ 𝐵
2		ax-frege58b 39035	. . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3		sbsbc 3666	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	sylib 210	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
5		dfsbcq 3664	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	4, 5	syl5ib 236	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
7	6	vtocleg 3496	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
8	1, 7	ax-mp 5	1 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1656   = wceq 1658  [wsb 2069   ∈ wcel 2166  [wsbc 3662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-ext 2803  ax-frege58b 39035

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663

This theorem is referenced by:  frege59c  39056  frege60c  39057  frege61c  39058  frege62c  39059  frege67c  39064  frege72  39069  frege118  39115  frege120  39117