Theorem hbexd 1694

Description: Deduction form of bound-variable hypothesis builder hbex 1636. (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 1611	. 2 ⊢ (𝜑 → (∃𝑦𝜓 → ∃𝑦∀𝑥𝜓))
4		19.12 1665	. 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓)
5	3, 4	syl6 33	1 ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1351  ∃wex 1492

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)