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Theorem albidh 1385
Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-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 1374 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 albi 1373 . 2 (∀𝑥(𝜓𝜒) → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
53, 4syl 14 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  nfbidf  1448  albid  1522  dral2  1635  ax11v2  1717  albidv  1721  equs5or  1727  sbal2  1914  eubidh  1922
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