Theorem 19.36vv 45121

Description: Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
19.36vv	⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem 19.36vv

Step	Hyp	Ref	Expression
1		19.36v 2022	. . 3 ⊢ (∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦𝜑 → 𝜓))
2	1	exbii 1877	. 2 ⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ ∃𝑥(∀𝑦𝜑 → 𝜓))
3		19.36v 2022	. 2 ⊢ (∃𝑥(∀𝑦𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓))
4	2, 3	bitri 278	1 ⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996

This proof depends on definitions:  df-bi 210  df-ex 1809

This theorem is used by: (None)