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Theorem albid 1522
 Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
Hypotheses
Ref Expression
albid.1 𝑥𝜑
albid.2 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
albid (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Proof of Theorem albid
StepHypRef Expression
1 albid.1 . . 3 𝑥𝜑
21nfri 1428 . 2 (𝜑 → ∀𝑥𝜑)
3 albid.2 . 2 (𝜑 → (𝜓𝜒))
42, 3albidh 1385 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  ∀wal 1257  Ⅎwnf 1365 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  ax-4 1416 This theorem depends on definitions:  df-bi 114  df-nf 1366 This theorem is referenced by:  alexdc  1526  19.32dc  1585  eubid  1923  ralbida  2337  raleqf  2518  intab  3672  bdsepnft  10394  strcollnft  10496  sscoll2  10500
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