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Theorem mpteq12dva 3838
 Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpteq12dv.1 (𝜑𝐴 = 𝐶)
mpteq12dva.2 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12dva (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem mpteq12dva
StepHypRef Expression
1 mpteq12dv.1 . . 3 (𝜑𝐴 = 𝐶)
21alrimiv 1754 . 2 (𝜑 → ∀𝑥 𝐴 = 𝐶)
3 mpteq12dva.2 . . 3 ((𝜑𝑥𝐴) → 𝐵 = 𝐷)
43ralrimiva 2392 . 2 (𝜑 → ∀𝑥𝐴 𝐵 = 𝐷)
5 mpteq12f 3837 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
62, 4, 5syl2anc 391 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ↦ cmpt 3818 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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-opab 3819  df-mpt 3820 This theorem is referenced by:  mpteq12dv  3839
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