Theorem 19.36i 1683

Description: Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Hypotheses

Ref	Expression
19.36i.1	⊢ Ⅎ𝑥𝜓
19.36i.2	⊢ ∃𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
19.36i	⊢ (∀𝑥𝜑 → 𝜓)

Proof of Theorem 19.36i

Step	Hyp	Ref	Expression
1		19.36i.2	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
2	1	19.35i 1636	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
3		19.36i.1	. . 3 ⊢ Ⅎ𝑥𝜓
4		id 19	. . 3 ⊢ (𝜓 → 𝜓)
5	3, 4	exlimi 1605	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
6	2, 5	syl 14	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1362  Ⅎwnf 1471  ∃wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  spimfv  1710  19.36aiv  1913  vtoclf  2805