Theorem ralnex2OLD 3260

Description: Obsolete version of ralnex2 3259 as of 18-May-2023. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ralnex2OLD	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Proof of Theorem ralnex2OLD

Step	Hyp	Ref	Expression
1		notnotb 317	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑)
2		notnotb 317	. . . 4 ⊢ (𝜑 ↔ ¬ ¬ 𝜑)
3	2	2rexbii 3247	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ ¬ 𝜑)
4		rexnal2 3257	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑)
5	3, 4	bitr2i 278	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
6	1, 5	xchbinx 336	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wral 3137  ∃wrex 3138

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 209  df-an 399  df-ex 1780  df-ral 3142  df-rex 3143

This theorem is referenced by: (None)