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Theorem mpt2eq12 5662
 Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpt2eq12 ((A = C B = D) → (x A, y B E) = (x C, y D E))
Distinct variable groups:   x,y,A   x,B,y   x,C,y   x,D,y
Allowed substitution hints:   E(x,y)

Proof of Theorem mpt2eq12
StepHypRef Expression
1 eqid 2353 . . . . 5 E = E
21rgenw 2681 . . . 4 y B E = E
32jctr 526 . . 3 (B = D → (B = D y B E = E))
43ralrimivw 2698 . 2 (B = Dx A (B = D y B E = E))
5 mpt2eq123 5661 . 2 ((A = C x A (B = D y B E = E)) → (x A, y B E) = (x C, y D E))
64, 5sylan2 460 1 ((A = C B = D) → (x A, y B E) = (x C, y D E))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  ∀wral 2614   ↦ cmpt2 5653 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-nfc 2478  df-ral 2619  df-oprab 5528  df-mpt2 5654 This theorem is referenced by: (None)
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