Theorem sbc5 3330

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 3315	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		exsimpl 1724	. . 3 |- ( E. x ( x = A /\ ph ) -> E. x x = A )
3		isset 3091	. . 3 |- ( A e. _V <-> E. x x = A )
4	2, 3	sylibr 215	. 2 |- ( E. x ( x = A /\ ph ) -> A e. _V )
5		dfsbcq2 3308	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		eqeq2 2444	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
7	6	anbi1d 709	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
8	7	exbidv 1761	. . 3 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
9		sb5 2226	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
10	5, 8, 9	vtoclbg 3146	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) )
11	1, 4, 10	pm5.21nii 354	1 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1659   [wsb 1789   e. wcel 1870   _Vcvv 3087   [.wsbc 3305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055  ax-ext 2407

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089  df-sbc 3306

This theorem is referenced by:  sbc6g  3331  sbc7  3333  sbciegft  3336  sbccomlem  3376  csb2  3403  rexsns  4035  rexsnsOLD  4036  iunxsngf  28011  sbccom2lem  32068  pm13.192  36398  pm13.195  36401  2sbc5g  36404  iotasbc  36407  pm14.122b  36411  iotasbc5  36419