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Theorem cdeqab 2814
 Description: Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
cdeqnot.1 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cdeqab CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem cdeqab
StepHypRef Expression
1 cdeqnot.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
21cdeqri 2810 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
32abbidv 2200 . 2 (𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
43cdeqi 2809 1 CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
 Colors of variables: wff set class Syntax hints:   ↔ wb 103   = wceq 1285  {cab 2069  CondEqwcdeq 2807 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-cdeq 2808 This theorem is referenced by: (None)
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