Theorem sbc5 3304

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 3289	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		exsimpl 1740	. . 3 |- ( E. x ( x = A /\ ph ) -> E. x x = A )
3		isset 3061	. . 3 |- ( A e. _V <-> E. x x = A )
4	2, 3	sylibr 217	. 2 |- ( E. x ( x = A /\ ph ) -> A e. _V )
5		dfsbcq2 3282	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		eqeq2 2473	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
7	6	anbi1d 716	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
8	7	exbidv 1779	. . 3 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
9		sb5 2270	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
10	5, 8, 9	vtoclbg 3120	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) )
11	1, 4, 10	pm5.21nii 359	1 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1455   E.wex 1674   [wsb 1808   e. wcel 1898   _Vcvv 3057   [.wsbc 3279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-v 3059  df-sbc 3280

This theorem is referenced by:  sbc6g  3305  sbc7  3307  sbciegft  3310  sbccomlem  3350  csb2  3377  rexsns  4016  rexsnsOLD  4017  iunxsngf  28227  sbccom2lem  32410  pm13.192  36806  pm13.195  36809  2sbc5g  36812  iotasbc  36815  pm14.122b  36819  iotasbc5  36827