Theorem sbss 3533

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 2742	. 2 |- y e. _V
2		sbequ 1840	. 2 |- ( z = y -> ( [ z / x ] x C_ A <-> [ y / x ] x C_ A ) )
3		sseq1 3180	. 2 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4		nfv 1528	. . 3 |- F/ x z C_ A
5		sseq1 3180	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
6	4, 5	sbie 1791	. 2 |- ( [ z / x ] x C_ A <-> z C_ A )
7	1, 2, 3, 6	vtoclb 2796	1 |- ( [ y / x ] x C_ A <-> y C_ A )

Syntax hints:   <-> wb 105   [wsb 1762   C_ wss 3131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2741  df-in 3137  df-ss 3144

This theorem is referenced by: (None)