Theorem sbss 3476

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 2692	. 2 |- y e. _V
2		sbequ 1813	. 2 |- ( z = y -> ( [ z / x ] x C_ A <-> [ y / x ] x C_ A ) )
3		sseq1 3125	. 2 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4		nfv 1509	. . 3 |- F/ x z C_ A
5		sseq1 3125	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
6	4, 5	sbie 1765	. 2 |- ( [ z / x ] x C_ A <-> z C_ A )
7	1, 2, 3, 6	vtoclb 2746	1 |- ( [ y / x ] x C_ A <-> y C_ A )

Syntax hints:   <-> wb 104   [wsb 1736   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691  df-in 3082  df-ss 3089

This theorem is referenced by: (None)