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Mirrors > Home > MPE Home > Th. List > rexnal2 | Structured version Visualization version GIF version |
Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
rexnal2 | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexnal 3090 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 𝜑) | |
2 | 1 | rexbii 3084 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 𝜑) |
3 | rexnal 3090 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wral 3051 ∃wrex 3060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-ral 3052 df-rex 3061 |
This theorem is referenced by: rexnal3 3126 2nreu 4439 nf1const 7307 cat1 18112 isnsgrp 18709 nn0prpw 36046 smprngopr 37764 aks6d1c6lem3 41882 fimgmcyc 42222 clsk1independent 43748 ichnreuop 47078 |
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