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Theorem unceq 36084
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 6846 . . . 4 (𝐴 = 𝐵 → (𝐴𝑥) = (𝐵𝑥))
21breqd 5121 . . 3 (𝐴 = 𝐵 → (𝑦(𝐴𝑥)𝑧𝑦(𝐵𝑥)𝑧))
32oprabbidv 7428 . 2 (𝐴 = 𝐵 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧})
4 df-unc 8204 . 2 uncurry 𝐴 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐴𝑥)𝑧}
5 df-unc 8204 . 2 uncurry 𝐵 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑦(𝐵𝑥)𝑧}
63, 4, 53eqtr4g 2802 1 (𝐴 = 𝐵 → uncurry 𝐴 = uncurry 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542   class class class wbr 5110  cfv 6501  {coprab 7363  uncurry cunc 8202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-in 3922  df-ss 3932  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-oprab 7366  df-unc 8204
This theorem is referenced by: (None)
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