Theorem hbexd 1708

Description: Deduction form of bound-variable hypothesis builder hbex 1650. (Contributed by NM, 2-Jan-2002.)

Hypotheses

Ref	Expression
hbexd.1	⊢ (𝜑 → ∀𝑦𝜑)
hbexd.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
hbexd	⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓))

Proof of Theorem hbexd

Step	Hyp	Ref	Expression
1		hbexd.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2		hbexd.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	1, 2	eximdh 1625	. 2 ⊢ (𝜑 → (∃𝑦𝜓 → ∃𝑦∀𝑥𝜓))
4		19.12 1679	. 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓)
5	3, 4	syl6 33	1 ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1362  ∃wex 1506

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)