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Theorem hbbid 1537
Description: Deduction form of bound-variable hypothesis builder hbbi 1510. (Contributed by NM, 1-Jan-2002.)
Hypotheses
Ref Expression
hbbid.1 (𝜑 → ∀𝑥𝜑)
hbbid.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
hbbid.3 (𝜑 → (𝜒 → ∀𝑥𝜒))
Assertion
Ref Expression
hbbid (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))

Proof of Theorem hbbid
StepHypRef Expression
1 hbbid.1 . . . 4 (𝜑 → ∀𝑥𝜑)
2 hbbid.2 . . . 4 (𝜑 → (𝜓 → ∀𝑥𝜓))
3 hbbid.3 . . . 4 (𝜑 → (𝜒 → ∀𝑥𝜒))
41, 2, 3hbimd 1535 . . 3 (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
51, 3, 2hbimd 1535 . . 3 (𝜑 → ((𝜒𝜓) → ∀𝑥(𝜒𝜓)))
64, 5anim12d 331 . 2 (𝜑 → (((𝜓𝜒) ∧ (𝜒𝜓)) → (∀𝑥(𝜓𝜒) ∧ ∀𝑥(𝜒𝜓))))
7 dfbi2 383 . 2 ((𝜓𝜒) ↔ ((𝜓𝜒) ∧ (𝜒𝜓)))
8 albiim 1446 . 2 (∀𝑥(𝜓𝜒) ↔ (∀𝑥(𝜓𝜒) ∧ ∀𝑥(𝜒𝜓)))
96, 7, 83imtr4g 204 1 (𝜑 → ((𝜓𝜒) → ∀𝑥(𝜓𝜒)))
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  ax-i5r 1498
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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