Theorem sbss 4084

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 3203	. 2 |- y e. _V
2		sbequ 2376	. 2 |- ( z = y -> ( [ z / x ] x C_ A <-> [ y / x ] x C_ A ) )
3		sseq1 3626	. 2 |- ( z = y -> ( z C_ A <-> y C_ A ) )
4		nfv 1843	. . 3 |- F/ x z C_ A
5		sseq1 3626	. . 3 |- ( x = z -> ( x C_ A <-> z C_ A ) )
6	4, 5	sbie 2408	. 2 |- ( [ z / x ] x C_ A <-> z C_ A )
7	1, 2, 3, 6	vtoclb 3263	1 |- ( [ y / x ] x C_ A <-> y C_ A )

Syntax hints:   <-> wb 196   [wsb 1880   C_ wss 3574

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-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by: (None)