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Theorem cdeqal 3711
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
cdeqal 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 cdeqal
StepHypRef Expression
1 cdeqnot.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
21cdeqri 3708 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
32albidv 1921 . 2 (𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
43cdeqi 3707 1 CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  CondEqwcdeq 3705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911
This theorem depends on definitions:  df-bi 210  df-cdeq 3706
This theorem is referenced by: (None)
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