Theorem frege58c 44374

Description: Principle related to sp 2195. 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 44354	. . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3		sbsbc 3727	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	sylib 219	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
5		dfsbcq 3725	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	4, 5	imbitrid 245	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
7	6	vtocleg 3499	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
8	1, 7	ax-mp 5	1 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1545   = wceq 1547  [wsb 2073   ∈ wcel 2119  [wsbc 3723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-frege58b 44354

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-sbc 3724

This theorem is referenced by:  frege59c  44375  frege60c  44376  frege61c  44377  frege62c  44378  frege67c  44383  frege72  44388  frege118  44434  frege120  44436