Theorem csbiedf 3173

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbiedf.1	|- F/xph
csbiedf.2	|- (ph -> F/_xC)
csbiedf.3	|- (ph -> A e. V)
csbiedf.4	|- ((ph /\ x = A) -> B = C)

Assertion

Ref	Expression
csbiedf	|- (ph -> [_A / x]_B = C)

Distinct variable group:   x,A

Allowed substitution hints:   ph(x)   B(x)   C(x)   V(x)

Proof of Theorem csbiedf

Step	Hyp	Ref	Expression
1		csbiedf.1	. . 3 |- F/xph
2		csbiedf.4	. . . 4 |- ((ph /\ x = A) -> B = C)
3	2	ex 423	. . 3 |- (ph -> (x = A -> B = C))
4	1, 3	alrimi 1765	. 2 |- (ph -> A.x(x = A -> B = C))
5		csbiedf.3	. . 3 |- (ph -> A e. V)
6		csbiedf.2	. . 3 |- (ph -> F/_xC)
7		csbiebt 3172	. . 3 |- ((A e. V /\ F/_xC) -> (A.x(x = A -> B = C) <-> [_A / x]_B = C))
8	5, 6, 7	syl2anc 642	. 2 |- (ph -> (A.x(x = A -> B = C) <-> [_A / x]_B = C))
9	4, 8	mpbid 201	1 |- (ph -> [_A / x]_B = C)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2476  [_csb 3136

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  df-csb 3137

This theorem is referenced by:  csbied  3178  csbie2t  3180