Theorem 19.23vv 1892

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 1891	. . 3 ⊢ (∀y(φ → ψ) ↔ (∃yφ → ψ))
2	1	albii 1566	. 2 ⊢ (∀x∀y(φ → ψ) ↔ ∀x(∃yφ → ψ))
3		19.23v 1891	. 2 ⊢ (∀x(∃yφ → ψ) ↔ (∃x∃yφ → ψ))
4	2, 3	bitri 240	1 ⊢ (∀x∀y(φ → ψ) ↔ (∃x∃yφ → ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This proof depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is used by:  ssrelk  4212  eqrelk  4213  sikexlem  4296  insklem  4305  raliunxp  4824  ssopr  4847