Theorem frege58c 38215

Description: Principle related to sp 2053. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege58c.a	|- A e. B

Assertion

Ref	Expression
frege58c	|- ( A. x ph -> [. A / x ]. ph )

Proof of Theorem frege58c

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege58c.a	. 2 |- A e. B
2		ax-frege58b 38195	. . . . 5 |- ( A. x ph -> [ y / x ] ph )
3		sbsbc 3439	. . . . 5 |- ( [ y / x ] ph <-> [. y / x ]. ph )
4	2, 3	sylib 208	. . . 4 |- ( A. x ph -> [. y / x ]. ph )
5		dfsbcq 3437	. . . 4 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
6	4, 5	syl5ib 234	. . 3 |- ( y = A -> ( A. x ph -> [. A / x ]. ph ) )
7	6	vtocleg 3279	. 2 |- ( A e. B -> ( A. x ph -> [. A / x ]. ph ) )
8	1, 7	ax-mp 5	1 |- ( A. x ph -> [. A / x ]. ph )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   [wsb 1880   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  frege59c  38216  frege60c  38217  frege61c  38218  frege62c  38219  frege67c  38224  frege72  38229  frege118  38275  frege120  38277