Theorem 19.23vv 1813

Description: Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
19.23vv	⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.23vv

Step	Hyp	Ref	Expression
1		19.23v 1812	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (∃𝑦𝜑 → 𝜓))
2	1	albii 1405	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(∃𝑦𝜑 → 𝜓))
3		19.23v 1812	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))
4	2, 3	bitri 183	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1288  ∃wex 1427

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie2 1429  ax-17 1465

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ssrel  4539  ssrelrel  4551  raliunxp  4590