Theorem sbciedf 3081

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	|- (ph -> A e. V)
sbcied.2	|- ((ph /\ x = A) -> (ps <-> ch))
sbciedf.3	|- F/xph
sbciedf.4	|- (ph -> F/xch)

Assertion

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

Distinct variable group:   x,A

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

Proof of Theorem sbciedf

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 |- (ph -> A e. V)
2		sbciedf.4	. 2 |- (ph -> F/xch)
3		sbciedf.3	. . 3 |- F/xph
4		sbcied.2	. . . 4 |- ((ph /\ x = A) -> (ps <-> ch))
5	4	ex 423	. . 3 |- (ph -> (x = A -> (ps <-> ch)))
6	3, 5	alrimi 1765	. 2 |- (ph -> A.x(x = A -> (ps <-> ch)))
7		sbciegft 3076	. 2 |- ((A e. V /\ F/xch /\ A.x(x = A -> (ps <-> ch))) -> ( [.A / x ].ps <-> ch))
8	1, 2, 6, 7	syl3anc 1182	1 |- (ph -> ( [.A / x ].ps <-> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   [.wsbc 3046

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 2478  df-v 2861  df-sbc 3047

This theorem is referenced by:  sbcied  3082  sbc2iegf  3112  csbiebt  3172  sbcnestgf  3183