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Theorem hbexd 1600
Description: Deduction form of bound-variable hypothesis builder hbex 1543. (Contributed by NM, 2-Jan-2002.)
Hypotheses
Ref Expression
hbexd.1 (𝜑 → ∀𝑦𝜑)
hbexd.2 (𝜑 → (𝜓 → ∀𝑥𝜓))
Assertion
Ref Expression
hbexd (𝜑 → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))

Proof of Theorem hbexd
StepHypRef Expression
1 hbexd.1 . . 3 (𝜑 → ∀𝑦𝜑)
2 hbexd.2 . . 3 (𝜑 → (𝜓 → ∀𝑥𝜓))
31, 2eximdh 1518 . 2 (𝜑 → (∃𝑦𝜓 → ∃𝑦𝑥𝜓))
4 19.12 1571 . 2 (∃𝑦𝑥𝜓 → ∀𝑥𝑦𝜓)
53, 4syl6 33 1 (𝜑 → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wex 1397
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-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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