Theorem sbss 3517

Description: Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
sbss	|- ( [ y / x ] x C_ A <-> y C_ A )

Distinct variable group:   x, A

Allowed substitution hint:   A( y)

Proof of Theorem sbss

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2729	. 2 |- y e. _V
2		sbequ 1828	. 2 |- ( z = y -> ( [ z / x ] x C_ A <-> [ y / x ] x C_ A ) )
3		sseq1 3165	. 2 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4		nfv 1516	. . 3 |- F/ x z C_ A
5		sseq1 3165	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
6	4, 5	sbie 1779	. 2 |- ( [ z / x ] x C_ A <-> z C_ A )
7	1, 2, 3, 6	vtoclb 2783	1 |- ( [ y / x ] x C_ A <-> y C_ A )

Syntax hints:   <-> wb 104   [wsb 1750   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728  df-in 3122  df-ss 3129

This theorem is referenced by: (None)