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Theorem 19.27h 1493
Description: Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
19.27h.1 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
19.27h (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.27h
StepHypRef Expression
1 19.26 1411 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2 19.27h.1 . . . 4 (𝜓 → ∀𝑥𝜓)
3219.3h 1486 . . 3 (∀𝑥𝜓𝜓)
43anbi2i 445 . 2 ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 182 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  aaanh  1519  19.27v  1821
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