Theorem sbcied 3083

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	|- (ph -> A e. V)
sbcied.2	|- ((ph /\ x = A) -> (ps <-> ch))

Assertion

Ref	Expression
sbcied	|- (ph -> ( [.A / x ].ps <-> ch))

Distinct variable groups:   x,A   ph,x   ch,x

Allowed substitution hints:   ps(x)   V(x)

Proof of Theorem sbcied

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 |- (ph -> A e. V)
2		sbcied.2	. 2 |- ((ph /\ x = A) -> (ps <-> ch))
3		nfv 1619	. 2 |- F/xph
4		nfvd 1620	. 2 |- (ph -> F/xch)
5	1, 2, 3, 4	sbciedf 3082	1 |- (ph -> ( [.A / x ].ps <-> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710   [.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-3an 936  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:  sbcied2  3084  sbc2iedv  3115  sbc3ie  3116  sbcralt  3119