NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  mpt2eq12 GIF version

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-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-nfc 2478  df-ral 2619  df-oprab 5528  df-mpt2 5654
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator