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Theorem mpt2eq12 5507
 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 B𝐸) = (x 𝐶, y 𝐷𝐸))
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 2037 . . . . 5 𝐸 = 𝐸
21rgenw 2370 . . . 4 y B 𝐸 = 𝐸
32jctr 298 . . 3 (B = 𝐷 → (B = 𝐷 y B 𝐸 = 𝐸))
43ralrimivw 2387 . 2 (B = 𝐷x A (B = 𝐷 y B 𝐸 = 𝐸))
5 mpt2eq123 5506 . 2 ((A = 𝐶 x A (B = 𝐷 y B 𝐸 = 𝐸)) → (x A, y B𝐸) = (x 𝐶, y 𝐷𝐸))
64, 5sylan2 270 1 ((A = 𝐶 B = 𝐷) → (x A, y B𝐸) = (x 𝐶, y 𝐷𝐸))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∀wral 2300   ↦ cmpt2 5457 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-oprab 5459  df-mpt2 5460 This theorem is referenced by:  iseqeq1  8874  iseqeq4  8877
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