Theorem sbc5 3071

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 3056	. 2 |- ( [.A / x ].ph -> A e. _V)
2		exsimpl 1592	. . 3 |- (E.x(x = A /\ ph) -> E.x x = A)
3		isset 2864	. . 3 |- (A e. _V <-> E.x x = A)
4	2, 3	sylibr 203	. 2 |- (E.x(x = A /\ ph) -> A e. _V)
5		dfsbcq2 3050	. . 3 |- (y = A -> ([y / x]ph <-> [.A / x ].ph))
6		eqeq2 2362	. . . . 5 |- (y = A -> (x = y <-> x = A))
7	6	anbi1d 685	. . . 4 |- (y = A -> ((x = y /\ ph) <-> (x = A /\ ph)))
8	7	exbidv 1626	. . 3 |- (y = A -> (E.x(x = y /\ ph) <-> E.x(x = A /\ ph)))
9		sb5 2100	. . 3 |- ([y / x]ph <-> E.x(x = y /\ ph))
10	5, 8, 9	vtoclbg 2916	. 2 |- (A e. _V -> ( [.A / x ].ph <-> E.x(x = A /\ ph)))
11	1, 4, 10	pm5.21nii 342	1 |- ( [.A / x ].ph <-> E.x(x = A /\ ph))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642  [wsb 1648   e. wcel 1710  _Vcvv 2860   [.wsbc 3047

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbc6g  3072  sbc7  3074  sbciegft  3077  sbccomlem  3117  csb2  3139  rexsns  3765