Theorem ralnex3 3147

Description: Relationship between three restricted universal and existential quantifiers. (Contributed by Thierry Arnoux, 12-Jul-2020.) (Proof shortened by Wolf Lammen, 18-May-2023.)

Assertion

Ref	Expression
ralnex3	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑)

Proof of Theorem ralnex3

Step	Hyp	Ref	Expression
1		ralnex 3092	. . 3 ⊢ (∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑧 ∈ 𝐶 𝜑)
2	1	2ralbii 3141	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ ∃𝑧 ∈ 𝐶 𝜑)
3		ralnex2 3146	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ ∃𝑧 ∈ 𝐶 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑)
4	2, 3	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wral 3080  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3081  df-rex 3091

This theorem is used by:  axtgupdim2  28960  usgrexmpl2trifr  49338