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Theorem unceq 37201
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 6895 . . . 4 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
21breqd 5160 . . 3 (𝐴 = 𝐵 → (𝑦(𝐴𝑥)𝑧𝑦(𝐵𝑥)𝑧))
32oprabbidv 7486 . 2 (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧})
4 df-unc 8274 . 2 uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧}
5 df-unc 8274 . 2 uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧}
63, 4, 53eqtr4g 2790 1 (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   class class class wbr 5149  cfv 6549  {coprab 7420  uncurry cunc 8272
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 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-ss 3961  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-oprab 7423  df-unc 8274
This theorem is referenced by: (None)
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