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Mirrors > Home > MPE Home > Th. List > rabeq0OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of rabeq0 3911 as of 16-Jul-2021. (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rabeq0OLD | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralnex 2975 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) | |
2 | rabn0OLD 3913 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝜑) | |
3 | 2 | necon1bbii 2831 | . 2 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = ∅) |
4 | 1, 3 | bitr2i 264 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∀wral 2896 ∃wrex 2897 {crab 2900 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: (None) |
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