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Theorem mpt2eq3dva 5511
 Description: Slightly more general equality inference for the maps to notation. (Contributed by NM, 17-Oct-2013.)
Hypothesis
Ref Expression
mpt2eq3dva.1 ((φ x A y B) → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2eq3dva (φ → (x A, y B𝐶) = (x A, y B𝐷))
Distinct variable groups:   φ,x   φ,y
Allowed substitution hints:   A(x,y)   B(x,y)   𝐶(x,y)   𝐷(x,y)

Proof of Theorem mpt2eq3dva
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 mpt2eq3dva.1 . . . . . 6 ((φ x A y B) → 𝐶 = 𝐷)
213expb 1104 . . . . 5 ((φ (x A y B)) → 𝐶 = 𝐷)
32eqeq2d 2048 . . . 4 ((φ (x A y B)) → (z = 𝐶z = 𝐷))
43pm5.32da 425 . . 3 (φ → (((x A y B) z = 𝐶) ↔ ((x A y B) z = 𝐷)))
54oprabbidv 5501 . 2 (φ → {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)} = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐷)})
6 df-mpt2 5460 . 2 (x A, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐶)}
7 df-mpt2 5460 . 2 (x A, y B𝐷) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) z = 𝐷)}
85, 6, 73eqtr4g 2094 1 (φ → (x A, y B𝐶) = (x A, y B𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  {coprab 5456   ↦ 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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-oprab 5459  df-mpt2 5460 This theorem is referenced by:  mpt2eq3ia  5512  ofeq  5656  fmpt2co  5779  iseqeq2  8895  iseqeq3  8896  iseqval  8900
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