Theorem 19.37vv 44384

Description: Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
19.37vv	⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.37vv

Step	Hyp	Ref	Expression
1		19.37v 1997	. . 3 ⊢ (∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑦𝜑))
2	1	exbii 1848	. 2 ⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ ∃𝑥(𝜓 → ∃𝑦𝜑))
3		19.37v 1997	. 2 ⊢ (∃𝑥(𝜓 → ∃𝑦𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))
4	2, 3	bitri 275	1 ⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by: (None)