MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cdeqal1 Structured version   Visualization version   GIF version

Theorem cdeqal1 3762
Description: Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
cdeqnot.1 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cdeqal1 CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cdeqal1
StepHypRef Expression
1 cdeqnot.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
21cdeqri 3757 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
32cbvalv 2393 . 2 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
43cdeqth 3758 1 CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1531  CondEqwcdeq 3754
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-11 2146  ax-12 2163  ax-13 2365
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-cdeq 3755
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator