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Mirrors > Home > MPE Home > Th. List > nfsb2ALT | Structured version Visualization version GIF version |
Description: Alternate version of nfsb2 2521. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.p3 | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Ref | Expression |
---|---|
nfsb2ALT | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfna1 2155 | . 2 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
2 | dfsb1.p3 | . . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
3 | 2 | hbsb2ALT 2598 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜃 → ∀𝑥𝜃)) |
4 | 1, 3 | nf5d 2291 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∃wex 1779 Ⅎwnf 1783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: nfsb4tALT 2603 |
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