Theorem sbc5 3361

Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)

Assertion

Ref	Expression
sbc5	|- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem sbc5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3346	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		exsimpl 1654	. . 3 |- ( E. x ( x = A /\ ph ) -> E. x x = A )
3		isset 3122	. . 3 |- ( A e. _V <-> E. x x = A )
4	2, 3	sylibr 212	. 2 |- ( E. x ( x = A /\ ph ) -> A e. _V )
5		dfsbcq2 3339	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		eqeq2 2482	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
7	6	anbi1d 704	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
8	7	exbidv 1690	. . 3 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
9		sb5 2157	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
10	5, 8, 9	vtoclbg 3177	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) )
11	1, 4, 10	pm5.21nii 353	1 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   [wsb 1711   e. wcel 1767   _Vcvv 3118   [.wsbc 3336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3120  df-sbc 3337

This theorem is referenced by:  sbc6g  3362  sbc7  3364  sbciegft  3367  sbccomlem  3415  csb2  3442  rexsns  4066  rexsnsOLD  4067  sbccom2lem  30457  pm13.192  31219  pm13.195  31222  2sbc5g  31225  iotasbc  31228  pm14.122b  31232  iotasbc5  31240