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Theorem mpt2eq12 5484
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 2018 . . . . 5 𝐸 = 𝐸
21rgenw 2350 . . . 4 y B 𝐸 = 𝐸
32jctr 298 . . 3 (B = 𝐷 → (B = 𝐷 y B 𝐸 = 𝐸))
43ralrimivw 2367 . 2 (B = 𝐷x A (B = 𝐷 y B 𝐸 = 𝐸))
5 mpt2eq123 5483 . 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 1226  wral 2280  cmpt2 5434
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-oprab 5436  df-mpt2 5437
This theorem is referenced by: (None)
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