Theorem 19.9hd 1640

Description: A deduction version of one direction of 19.9 1623. This is an older variation of this theorem; new proofs should use 19.9d 1639. (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 → (∃𝑥𝜑 → 𝜑))

Proof of Theorem 19.9hd

Step	Hyp	Ref	Expression
1		19.9hd.1	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2		19.9hd.2	. . 3 ⊢ (𝜓 → (𝜑 → ∀𝑥𝜑))
3	2	alimi 1431	. 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → ∀𝑥𝜑))
4		19.9ht 1620	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
5	1, 3, 4	3syl 17	1 ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1423  ax-gen 1425  ax-ie2 1470

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  sbiedh  1760  bj-sbimedh  12967