Theorem mpteq2dva 5668

Description: Slightly more general equality inference for the maps to notation. (Contributed by Scott Fenton, 25-Apr-2012.)

Hypothesis

Ref	Expression
mpteq2dva.1	|- ((ph /\ x e. A) -> B = C)

Assertion

Ref	Expression
mpteq2dva	|- (ph -> (x e. A |-> B) = (x e. A |-> C))

Distinct variable group:   ph,x

Allowed substitution hints:   A(x)   B(x)   C(x)

Proof of Theorem mpteq2dva

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/xph
2		mpteq2dva.1	. 2 |- ((ph /\ x e. A) -> B = C)
3	1, 2	mpteq2da 5667	1 |- (ph -> (x e. A |-> B) = (x e. A |-> C))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   |-> cmpt 5652

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2620  df-opab 4624  df-mpt 5653

This theorem is referenced by:  mpteq2dv  5669