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Theorem mpteq12df 5109
Description: An equality inference for the maps-to notation. Compare mpteq12dv 5112. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
mpteq12df.1 𝑥𝜑
mpteq12df.2 (𝜑𝐴 = 𝐶)
mpteq12df.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12df (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Proof of Theorem mpteq12df
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mpteq12df.1 . . 3 𝑥𝜑
2 nfv 1920 . . 3 𝑦𝜑
3 mpteq12df.2 . . . . 5 (𝜑𝐴 = 𝐶)
43eleq2d 2818 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐶))
5 mpteq12df.3 . . . . 5 (𝜑𝐵 = 𝐷)
65eqeq2d 2749 . . . 4 (𝜑 → (𝑦 = 𝐵𝑦 = 𝐷))
74, 6anbi12d 634 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐷)))
81, 2, 7opabbid 5092 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
9 df-mpt 5108 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
10 df-mpt 5108 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
118, 9, 103eqtr4g 2798 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wnf 1790  wcel 2113  {copab 5089  cmpt 5107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-opab 5090  df-mpt 5108
This theorem is referenced by:  mpteq12dv  5112  esumrnmpt2  31598  exrecfnlem  35162  smflimmpt  43866
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