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Theorem 19.28h 1524
Description: Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
19.28h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
19.28h (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.28h
StepHypRef Expression
1 19.26 1440 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.28h.1 . . . 4 (𝜑 → ∀𝑥𝜑)
3219.3h 1515 . . 3 (∀𝑥𝜑𝜑)
43anbi1i 451 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
51, 4bitri 183 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-4 1470
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  nfan1  1526  aaanh  1548  exan  1654  19.28v  1854
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