Theorem rexnal3 3254

Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.)

Assertion

Ref	Expression
rexnal3	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑)

Proof of Theorem rexnal3

Step	Hyp	Ref	Expression
1		rexnal 3203	. . 3 ⊢ (∃𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∀𝑧 ∈ 𝐶 𝜑)
2	1	2rexbii 3252	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ ∀𝑧 ∈ 𝐶 𝜑)
3		rexnal2 3253	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ ∀𝑧 ∈ 𝐶 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑)
4	2, 3	bitri 267	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wral 3117  ∃wrex 3118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-ral 3122  df-rex 3123

This theorem is referenced by:  ralnex3  3256