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Theorem albidh 1369
 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 1358 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
4 albi 1357 . 2 (∀𝑥(𝜓𝜒) → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
53, 4syl 14 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  nfbidf  1432  albid  1506  dral2  1619  ax11v2  1701  albidv  1705  equs5or  1711  sbal2  1898  eubidh  1906
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