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Theorem mpoeq3dva 5847
 Description: Slightly more general equality inference for the maps-to notation. (Contributed by NM, 17-Oct-2013.)
Hypothesis
Ref Expression
mpoeq3dva.1 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
Assertion
Ref Expression
mpoeq3dva (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
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 mpoeq3dva
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mpoeq3dva.1 . . . . . 6 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 = 𝐷)
213expb 1183 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶 = 𝐷)
32eqeq2d 2153 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑧 = 𝐶𝑧 = 𝐷))
43pm5.32da 448 . . 3 (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)))
54oprabbidv 5837 . 2 (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)})
6 df-mpo 5791 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
7 df-mpo 5791 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐷)}
85, 6, 73eqtr4g 2199 1 (𝜑 → (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑦𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 2112  {coprab 5787   ∈ cmpo 5788 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-oprab 5790  df-mpo 5791 This theorem is referenced by:  mpoeq3ia  5848  mpoeq3dv  5849  ofeq  5996  fmpoco  6125  mapxpen  6754  seqeq2  10282  seqeq3  10283  cnmpt2t  12537  cnmpt22  12538  cnmptcom  12542
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