Theorem frege58c 40287

Description: Principle related to sp 2182. 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 40267	. . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3		sbsbc 3776	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
4	2, 3	sylib 220	. . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
5		dfsbcq 3774	. . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
6	4, 5	syl5ib 246	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
7	6	vtocleg 3581	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
8	1, 7	ax-mp 5	1 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1535   = wceq 1537  [wsb 2069   ∈ wcel 2114  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793  ax-frege58b 40267

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3773

This theorem is referenced by:  frege59c  40288  frege60c  40289  frege61c  40290  frege62c  40291  frege67c  40296  frege72  40301  frege118  40347  frege120  40349