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Theorem a5i 1505
Description: Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
a5i.1 (∀𝑥𝜑𝜓)
Assertion
Ref Expression
a5i (∀𝑥𝜑 → ∀𝑥𝜓)

Proof of Theorem a5i
StepHypRef Expression
1 hba1 1503 . . 3 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
2 ax-5 1406 . . 3 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝑥𝜑 → ∀𝑥𝜓))
31, 2syl5 32 . 2 (∀𝑥(∀𝑥𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
4 a5i.1 . 2 (∀𝑥𝜑𝜓)
53, 4mpg 1410 1 (∀𝑥𝜑 → ∀𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1406  ax-gen 1408  ax-ial 1497
This theorem is referenced by:  hbae  1679  equveli  1715  hbsb2a  1760  hbsb2e  1761  aev  1766  dveeq2or  1770  hbsb2  1790  nfsb2or  1791  reu6  2844
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