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Theorem 19.9hd 1593
Description: A deduction version of one direction of 19.9 1576. This is an older variation of this theorem; new proofs should use 19.9d 1592. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Hypotheses
Ref Expression
19.9hd.1 (𝜓 → ∀𝑥𝜓)
19.9hd.2 (𝜓 → (𝜑 → ∀𝑥𝜑))
Assertion
Ref Expression
19.9hd (𝜓 → (∃𝑥𝜑𝜑))

Proof of Theorem 19.9hd
StepHypRef Expression
1 19.9hd.1 . 2 (𝜓 → ∀𝑥𝜓)
2 19.9hd.2 . . 3 (𝜓 → (𝜑 → ∀𝑥𝜑))
32alimi 1385 . 2 (∀𝑥𝜓 → ∀𝑥(𝜑 → ∀𝑥𝜑))
4 19.9ht 1573 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
51, 3, 43syl 17 1 (𝜓 → (∃𝑥𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-5 1377  ax-gen 1379  ax-ie2 1424
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  sbiedh  1712  bj-sbimedh  10842
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