Theorem alinexa 1578

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)

Assertion

Ref	Expression
alinexa	⊢ (∀x(φ → ¬ ψ) ↔ ¬ ∃x(φ ∧ ψ))

Proof of Theorem alinexa

Step	Hyp	Ref	Expression
1		imnan 411	. . 3 ⊢ ((φ → ¬ ψ) ↔ ¬ (φ ∧ ψ))
2	1	albii 1566	. 2 ⊢ (∀x(φ → ¬ ψ) ↔ ∀x ¬ (φ ∧ ψ))
3		alnex 1543	. 2 ⊢ (∀x ¬ (φ ∧ ψ) ↔ ¬ ∃x(φ ∧ ψ))
4	2, 3	bitri 240	1 ⊢ (∀x(φ → ¬ ψ) ↔ ¬ ∃x(φ ∧ ψ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is used by:  equs3  1644  axi11e  2332  ralnex  2625