Theorem 19.23vv 1946

Description: Theorem 19.23v 1945 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 1945	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (∃𝑦𝜑 → 𝜓))
2	1	albii 1822	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(∃𝑦𝜑 → 𝜓))
3		19.23v 1945	. 2 ⊢ (∀𝑥(∃𝑦𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))
4	2, 3	bitri 274	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  ssrel  5693  ssrelOLD  5694  ssrelrel  5706  raliunxp  5748  bnj1052  32955  bnj1030  32967