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Theorem albidh 1790
Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
Hypotheses
Ref Expression
albidh.1 (𝜑 → ∀𝑥𝜑)
albidh.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
albidh (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Proof of Theorem albidh
StepHypRef Expression
1 albidh.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 albidh.2 . . 3 (𝜑 → (𝜓𝜒))
31, 2alrimih 1748 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 albi 1743 . 2 (∀𝑥(𝜓𝜒) → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
53, 4syl 17 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197
This theorem is referenced by:  albidv  1846  albid  2088  albidOLD  2198  dral2-o  33730  ax12indalem  33745  ax12inda2ALT  33746  ax12inda  33748
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