Theorem nrexralim 3137

Description: Negation of a complex predicate calculus formula. (Contributed by FL, 31-Jul-2009.)

Assertion

Ref	Expression
nrexralim	⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ ¬ 𝜓))

Proof of Theorem nrexralim

Step	Hyp	Ref	Expression
1		rexanali 3102	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓))
2	1	ralbii 3093	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓))
3		ralnex 3072	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓))
4	2, 3	bitr2i 276	1 ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wral 3061  ∃wrex 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-ral 3062  df-rex 3071

This theorem is referenced by: (None)