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Theorem cdeqal1 3671
Description: Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
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 3666 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
32cbvalv 2331 . 2 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
43cdeqth 3667 1 CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wal 1505  CondEqwcdeq 3663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-11 2093  ax-12 2106  ax-13 2301
This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-cdeq 3664
This theorem is referenced by: (None)
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