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Theorem hbimd 1508
Description: Deduction form of bound-variable hypothesis builder hbim 1480. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
Hypotheses
Ref Expression
hbimd.1 (𝜑 → ∀𝑥𝜑)
hbimd.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
hbimd.3 (𝜑 → (𝜒 → ∀𝑥𝜒))
Assertion
Ref Expression
hbimd (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Proof of Theorem hbimd
StepHypRef Expression
1 hbimd.3 . . . 4 (𝜑 → (𝜒 → ∀𝑥𝜒))
21imim2d 53 . . 3 (𝜑 → ((𝜓𝜒) → (𝜓 → ∀𝑥𝜒)))
3 ax-4 1443 . . . . 5 (∀𝑥𝜓𝜓)
43imim1i 59 . . . 4 ((𝜓 → ∀𝑥𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒))
5 ax-i5r 1471 . . . 4 ((∀𝑥𝜓 → ∀𝑥𝜒) → ∀𝑥(∀𝑥𝜓𝜒))
64, 5syl 14 . . 3 ((𝜓 → ∀𝑥𝜒) → ∀𝑥(∀𝑥𝜓𝜒))
72, 6syl6 33 . 2 (𝜑 → ((𝜓𝜒) → ∀𝑥(∀𝑥𝜓𝜒)))
8 hbimd.1 . . 3 (𝜑 → ∀𝑥𝜑)
9 hbimd.2 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
109imim1d 74 . . 3 (𝜑 → ((∀𝑥𝜓𝜒) → (𝜓𝜒)))
118, 10alimdh 1399 . 2 (𝜑 → (∀𝑥(∀𝑥𝜓𝜒) → ∀𝑥(𝜓𝜒)))
127, 11syld 44 1 (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-5 1379  ax-gen 1381  ax-4 1443  ax-i5r 1471
This theorem is referenced by:  hbbid  1510  19.21ht  1516  equveli  1686  dvelimfALT2  1742
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