Theorem sbss 3659

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 2862	. 2 |- y e. _V
2		sbequ 2060	. 2 |- (z = y -> ([z / x]x C_ A <-> [y / x]x C_ A))
3		sseq1 3292	. 2 |- (z = y -> (z C_ A <-> y C_ A))
4		nfv 1619	. . 3 |- F/x z C_ A
5		sseq1 3292	. . 3 |- (x = z -> (x C_ A <-> z C_ A))
6	4, 5	sbie 2038	. 2 |- ([z / x]x C_ A <-> z C_ A)
7	1, 2, 3, 6	vtoclb 2912	1 |- ([y / x]x C_ A <-> y C_ A)

Syntax hints:   <-> wb 176  [wsb 1648   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by: (None)