Theorem rexnal2 3190

Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
rexnal2	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Proof of Theorem rexnal2

Step	Hyp	Ref	Expression
1		rexnal 3141	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 𝜑)
2	1	rexbii 3188	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 𝜑)
3		rexnal 3141	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)
4	2, 3	bitri 266	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 197  ∀wral 3055  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904

This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-ral 3060  df-rex 3061

This theorem is referenced by:  rexnal3  3191  ralnex2  3192  isnsgrp  17556  tgdim01  25693  nn0prpw  32693  smprngopr  34205  clsk1independent  38950