Theorem hbexd 1687

Description: Deduction form of bound-variable hypothesis builder hbex 1629. (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 1604	. 2 ⊢ (𝜑 → (∃𝑦𝜓 → ∃𝑦∀𝑥𝜓))
4		19.12 1658	. 2 ⊢ (∃𝑦∀𝑥𝜓 → ∀𝑥∃𝑦𝜓)
5	3, 4	syl6 33	1 ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓))

Syntax hints:   → wi 4  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)