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Theorem mpteq12d 4866
Description: An equality inference for the maps to notation. Compare mpteq12dv 4865. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
mpteq12d.1 𝑥𝜑
mpteq12d.3 (𝜑𝐴 = 𝐶)
mpteq12d.4 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
mpteq12d (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Proof of Theorem mpteq12d
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mpteq12d.1 . . 3 𝑥𝜑
2 nfv 1994 . . 3 𝑦𝜑
3 mpteq12d.3 . . . . 5 (𝜑𝐴 = 𝐶)
43eleq2d 2835 . . . 4 (𝜑 → (𝑥𝐴𝑥𝐶))
5 mpteq12d.4 . . . . 5 (𝜑𝐵 = 𝐷)
65eqeq2d 2780 . . . 4 (𝜑 → (𝑦 = 𝐵𝑦 = 𝐷))
74, 6anbi12d 608 . . 3 (𝜑 → ((𝑥𝐴𝑦 = 𝐵) ↔ (𝑥𝐶𝑦 = 𝐷)))
81, 2, 7opabbid 4847 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)})
9 df-mpt 4862 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
10 df-mpt 4862 . 2 (𝑥𝐶𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦 = 𝐷)}
118, 9, 103eqtr4g 2829 1 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wnf 1855  wcel 2144  {copab 4844  cmpt 4861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-opab 4845  df-mpt 4862
This theorem is referenced by:  esumrnmpt2  30464  smflimmpt  41530
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