Theorem bnj1468 30916

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1468.1	|- ( ps -> A. x ps )
bnj1468.2	|- ( x = A -> ( ph <-> ps ) )
bnj1468.3	|- ( y e. A -> A. x y e. A )

Assertion

Ref	Expression
bnj1468	|- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Distinct variable groups:   y, A   y, V   ph, y   ps, y   x, y

Allowed substitution hints:   ph( x)   ps( x)   A( x)   V( x)

Proof of Theorem bnj1468

Step	Hyp	Ref	Expression
1		sbcco 3458	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
2		ax-5 1839	. . 3 |- ( ps -> A. y ps )
3		bnj1468.3	. . . . . . . 8 |- ( y e. A -> A. x y e. A )
4	3	nfcii 2755	. . . . . . 7 |- F/_ x A
5	4	nfeq2 2780	. . . . . 6 |- F/ x y = A
6		nfsbc1v 3455	. . . . . . 7 |- F/ x [. y / x ]. ph
7		bnj1468.1	. . . . . . . 8 |- ( ps -> A. x ps )
8	7	nf5i 2024	. . . . . . 7 |- F/ x ps
9	6, 8	nfbi 1833	. . . . . 6 |- F/ x ( [. y / x ]. ph <-> ps )
10	5, 9	nfim 1825	. . . . 5 |- F/ x ( y = A -> ( [. y / x ]. ph <-> ps ) )
11	10	nf5ri 2065	. . . 4 |- ( ( y = A -> ( [. y / x ]. ph <-> ps ) ) -> A. x ( y = A -> ( [. y / x ]. ph <-> ps ) ) )
12		ax6ev 1890	. . . . 5 |- E. x x = y
13		eqeq1 2626	. . . . . . 7 |- ( x = y -> ( x = A <-> y = A ) )
14		bnj1468.2	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
15	13, 14	syl6bir 244	. . . . . 6 |- ( x = y -> ( y = A -> ( ph <-> ps ) ) )
16		sbceq1a 3446	. . . . . . 7 |- ( x = y -> ( ph <-> [. y / x ]. ph ) )
17	16	bibi1d 333	. . . . . 6 |- ( x = y -> ( ( ph <-> ps ) <-> ( [. y / x ]. ph <-> ps ) ) )
18	15, 17	sylibd 229	. . . . 5 |- ( x = y -> ( y = A -> ( [. y / x ]. ph <-> ps ) ) )
19	12, 18	bnj101 30789	. . . 4 |- E. x ( y = A -> ( [. y / x ]. ph <-> ps ) )
20	11, 19	bnj1131 30858	. . 3 |- ( y = A -> ( [. y / x ]. ph <-> ps ) )
21	2, 20	bnj1464 30914	. 2 |- ( A e. V -> ( [. A / y ]. [. y / x ]. ph <-> ps ) )
22	1, 21	syl5bbr 274	1 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj1463  31123