Theorem sbss 3329

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 2560	. 2 |- y e. _V
2		sbequ 1721	. 2 |- ( z = y -> ( [ z / x ] x C_ A <-> [ y / x ] x C_ A ) )
3		sseq1 2966	. 2 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4		nfv 1421	. . 3 |- F/ x z C_ A
5		sseq1 2966	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
6	4, 5	sbie 1674	. 2 |- ( [ z / x ] x C_ A <-> z C_ A )
7	1, 2, 3, 6	vtoclb 2611	1 |- ( [ y / x ] x C_ A <-> y C_ A )

Syntax hints:   <-> wb 98   [wsb 1645   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by: (None)