Theorem exanali 725

Description: A transformation of quantifiers and logical connectives.

Assertion

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

Proof of Theorem exanali

Step	Hyp	Ref	Expression
1		iman 205	. . . 4 ⊢ ((φ → ψ) ↔ ¬ (φ ∧ ¬ ψ))
2	1	bial 695	. . 3 ⊢ (∀x(φ → ψ) ↔ ∀x ¬ (φ ∧ ¬ ψ))
3		alnex 716	. . 3 ⊢ (∀x ¬ (φ ∧ ¬ ψ) ↔ ¬ ∃x(φ ∧ ¬ ψ))
4	2, 3	bitr 151	. 2 ⊢ (∀x(φ → ψ) ↔ ¬ ∃x(φ ∧ ¬ ψ))
5	4	bicon2i 194	1 ⊢ (∃x(φ ∧ ¬ ψ) ↔ ¬ ∀x(φ → ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  rexnal 1210  gencbval 1373  prlem934 3933  reclem2pr 3951

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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