Theorem frege58c 41418

Description: Principle related to sp 2178. 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 41398	. . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3		sbsbc 3715	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	sylib 217	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
5		dfsbcq 3713	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	4, 5	syl5ib 243	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
7	6	vtocleg 3511	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
8	1, 7	ax-mp 5	1 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1537   = wceq 1539  [wsb 2068   ∈ 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-frege58b 41398

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-sbc 3712

This theorem is referenced by:  frege59c  41419  frege60c  41420  frege61c  41421  frege62c  41422  frege67c  41427  frege72  41432  frege118  41478  frege120  41480