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Mirrors > Home > NFE Home > Th. List > rexanali | GIF version |
Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) |
Ref | Expression |
---|---|
rexanali | ⊢ (∃x ∈ A (φ ∧ ¬ ψ) ↔ ¬ ∀x ∈ A (φ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | annim 414 | . . 3 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ)) | |
2 | 1 | rexbii 2639 | . 2 ⊢ (∃x ∈ A (φ ∧ ¬ ψ) ↔ ∃x ∈ A ¬ (φ → ψ)) |
3 | rexnal 2625 | . 2 ⊢ (∃x ∈ A ¬ (φ → ψ) ↔ ¬ ∀x ∈ A (φ → ψ)) | |
4 | 2, 3 | bitri 240 | 1 ⊢ (∃x ∈ A (φ ∧ ¬ ψ) ↔ ¬ ∀x ∈ A (φ → ψ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 ∀wral 2614 ∃wrex 2615 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-ral 2619 df-rex 2620 |
This theorem is referenced by: transex 5910 antisymex 5912 foundex 5914 extex 5915 symex 5916 nclennlem1 6248 nchoicelem16 6304 |
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