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Mirrors > Home > MPE Home > Th. List > ralinexa | Structured version Visualization version GIF version |
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) |
Ref | Expression |
---|---|
ralinexa | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imnan 400 | . . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
2 | 1 | ralbii 3096 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓)) |
3 | ralnex 3075 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∀wral 3064 ∃wrex 3073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-ral 3065 df-rex 3074 |
This theorem is referenced by: soseq 8090 kmlem7 10091 kmlem13 10097 lspsncv0 20605 ntreq0 22426 lhop1lem 25375 nogt01o 27042 ltrnnid 38590 |
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