Theorem 19.37vv 45128

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 2030	. . 3 ⊢ (∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑦𝜑))
2	1	exbii 1881	. 2 ⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ ∃𝑥(𝜓 → ∃𝑦𝜑))
3		19.37v 2030	. 2 ⊢ (∃𝑥(𝜓 → ∃𝑦𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))
4	2, 3	bitri 278	1 ⊢ (∃𝑥∃𝑦(𝜓 → 𝜑) ↔ (𝜓 → ∃𝑥∃𝑦𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by: (None)