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Theorem unceq 36968
Description: Equality theorem for uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)
Assertion
Ref Expression
unceq (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)

Proof of Theorem unceq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 6881 . . . 4 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
21breqd 5150 . . 3 (𝐴 = 𝐵 → (𝑦(𝐴𝑥)𝑧𝑦(𝐵𝑥)𝑧))
32oprabbidv 7468 . 2 (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧})
4 df-unc 8249 . 2 uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧}
5 df-unc 8249 . 2 uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧}
63, 4, 53eqtr4g 2789 1 (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   class class class wbr 5139  cfv 6534  {coprab 7403  uncurry cunc 8247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-oprab 7406  df-unc 8249
This theorem is referenced by: (None)
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