MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdeqab Structured version   Visualization version   GIF version

Theorem cdeqab 3683
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 3679 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
32abbidv 2807 . 2 (𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
43cdeqi 3678 1 CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  {cab 2714  CondEqwcdeq 3676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-cdeq 3677
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator