Theorem sbc5 3460

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 3445	. 2 |- ( [. A / x ]. ph -> A e. _V )
2		exsimpl 1795	. . 3 |- ( E. x ( x = A /\ ph ) -> E. x x = A )
3		isset 3207	. . 3 |- ( A e. _V <-> E. x x = A )
4	2, 3	sylibr 224	. 2 |- ( E. x ( x = A /\ ph ) -> A e. _V )
5		dfsbcq2 3438	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
6		eqeq2 2633	. . . . 5 |- ( y = A -> ( x = y <-> x = A ) )
7	6	anbi1d 741	. . . 4 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
8	7	exbidv 1850	. . 3 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
9		sb5 2430	. . 3 |- ( [ y / x ] ph <-> E. x ( x = y /\ ph ) )
10	5, 8, 9	vtoclbg 3267	. 2 |- ( A e. _V -> ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) ) )
11	1, 4, 10	pm5.21nii 368	1 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

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-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-sbc 3436

This theorem is referenced by:  sbc6g  3461  sbc7  3463  sbciegft  3466  sbccomlem  3508  csb2  3535  rexsns  4217  sbccom2lem  33929  pm13.192  38611  pm13.195  38614  2sbc5g  38617  iotasbc  38620  pm14.122b  38624  iotasbc5  38632