Theorem rexanali 1687

Description: A transformation of restricted quantifiers and logical connectives.

Assertion

Ref	Expression
rexanali	⊢ (∃x ∈ A (φ ⋀ ¬ ψ) ↔ ¬ ∀x ∈ A (φ → ψ))

Proof of Theorem rexanali

Step	Hyp	Ref	Expression
1		annim 238	. . 3 ⊢ ((φ ⋀ ¬ ψ) ↔ ¬ (φ → ψ))
2	1	rexbii 1671	. 2 ⊢ (∃x ∈ A (φ ⋀ ¬ ψ) ↔ ∃x ∈ A ¬ (φ → ψ))
3		rexnal 1657	. 2 ⊢ (∃x ∈ A ¬ (φ → ψ) ↔ ¬ ∀x ∈ A (φ → ψ))
4	2, 3	bitr 173	1 ⊢ (∃x ∈ A (φ ⋀ ¬ ψ) ↔ ¬ ∀x ∈ A (φ → ψ))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wral 1648  ∃wrex 1649

This theorem is referenced by:  supxrre 6085  qsqueeze 6281  climrecl 7110  climge0 7112  elcls 7701

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-ral 1652  df-rex 1653

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