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Theorem r19.28av 2466
Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.28av ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.28av
StepHypRef Expression
1 r19.27av 2465 . 2 ((∀𝑥𝐴 𝜓𝜑) → ∀𝑥𝐴 (𝜓𝜑))
2 ancom 257 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜓𝜑))
3 ancom 257 . . 3 ((𝜑𝜓) ↔ (𝜓𝜑))
43ralbii 2347 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜓𝜑))
51, 2, 43imtr4i 194 1 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wral 2323
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  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-ral 2328
This theorem is referenced by:  rr19.28v  2706  fununi  4995
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