Theorem exanali 1585

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)

Assertion

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

Proof of Theorem exanali

Step	Hyp	Ref	Expression
1		annim 414	. . 3 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ))
2	1	exbii 1582	. 2 ⊢ (∃x(φ ∧ ¬ ψ) ↔ ∃x ¬ (φ → ψ))
3		exnal 1574	. 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:  ax11indn  2195  rexnal  2626  gencbval  2904  nss  3330  ssfin  4471  ncfinlowerlem1  4483  spfinex  4538  nfunv  5139  funsex  5829  fnfullfunlem1  5857  foundex  5915  fnfreclem1  6318