Theorem bj-csbsnlem 32898

Description: Lemma for bj-csbsn 32899 (in this lemma, x cannot occur in A). (Contributed by BJ, 6-Oct-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-csbsnlem	|- [_ A / x ]_ { x } = { A }

Distinct variable group:   x, A

Proof of Theorem bj-csbsnlem

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abid 2610	. . . 4 |- ( y e. { y | [. A / x ]. y e. { x } } <-> [. A / x ]. y e. { x } )
2		df-sbc 3436	. . . 4 |- ( [. A / x ]. y e. { x } <-> A e. { x | y e. { x } } )
3		clelab 2748	. . . . 5 |- ( A e. { x | y e. { x } } <-> E. x ( x = A /\ y e. { x } ) )
4		velsn 4193	. . . . . . 7 |- ( y e. { x } <-> y = x )
5	4	anbi2i 730	. . . . . 6 |- ( ( x = A /\ y e. { x } ) <-> ( x = A /\ y = x ) )
6	5	exbii 1774	. . . . 5 |- ( E. x ( x = A /\ y e. { x } ) <-> E. x ( x = A /\ y = x ) )
7		eqeq2 2633	. . . . . . . 8 |- ( x = A -> ( y = x <-> y = A ) )
8	7	pm5.32i 669	. . . . . . 7 |- ( ( x = A /\ y = x ) <-> ( x = A /\ y = A ) )
9	8	exbii 1774	. . . . . 6 |- ( E. x ( x = A /\ y = x ) <-> E. x ( x = A /\ y = A ) )
10		19.41v 1914	. . . . . 6 |- ( E. x ( x = A /\ y = A ) <-> ( E. x x = A /\ y = A ) )
11		simpr 477	. . . . . . 7 |- ( ( E. x x = A /\ y = A ) -> y = A )
12		eqvisset 3211	. . . . . . . . 9 |- ( y = A -> A e. _V )
13		elisset 3215	. . . . . . . . 9 |- ( A e. _V -> E. x x = A )
14	12, 13	syl 17	. . . . . . . 8 |- ( y = A -> E. x x = A )
15	14	ancri 575	. . . . . . 7 |- ( y = A -> ( E. x x = A /\ y = A ) )
16	11, 15	impbii 199	. . . . . 6 |- ( ( E. x x = A /\ y = A ) <-> y = A )
17	9, 10, 16	3bitri 286	. . . . 5 |- ( E. x ( x = A /\ y = x ) <-> y = A )
18	3, 6, 17	3bitri 286	. . . 4 |- ( A e. { x | y e. { x } } <-> y = A )
19	1, 2, 18	3bitri 286	. . 3 |- ( y e. { y | [. A / x ]. y e. { x } } <-> y = A )
20		df-csb 3534	. . . 4 |- [_ A / x ]_ { x } = { y | [. A / x ]. y e. { x } }
21	20	eleq2i 2693	. . 3 |- ( y e. [_ A / x ]_ { x } <-> y e. { y | [. A / x ]. y e. { x } } )
22		velsn 4193	. . 3 |- ( y e. { A } <-> y = A )
23	19, 21, 22	3bitr4i 292	. 2 |- ( y e. [_ A / x ]_ { x } <-> y e. { A } )
24	23	eqriv 2619	1 |- [_ A / x ]_ { x } = { A }

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   [.wsbc 3435   [_csb 3533   {csn 4177

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-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  df-csb 3534  df-sn 4178

This theorem is referenced by:  bj-csbsn  32899