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Theorem mpteq12dv 5656
 Description: An equality inference for the maps to notation. (Contributed by set.mm contributors, 24-Aug-2011.) (Revised by set.mm contributors, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq12dv.1 (φA = C)
mpteq12dv.2 (φB = D)
Assertion
Ref Expression
mpteq12dv (φ → (x A B) = (x C D))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   C(x)   D(x)

Proof of Theorem mpteq12dv
StepHypRef Expression
1 mpteq12dv.1 . . 3 (φA = C)
21alrimiv 1631 . 2 (φx A = C)
3 mpteq12dv.2 . . 3 (φB = D)
43ralrimivw 2698 . 2 (φx A B = D)
5 mpteq12f 5655 . 2 ((x A = C x A B = D) → (x A B) = (x C D))
62, 4, 5syl2anc 642 1 (φ → (x A B) = (x C D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642  ∀wral 2614   ↦ cmpt 5651 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2619  df-opab 4623  df-mpt 5652 This theorem is referenced by:  mpteq12i  5665
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