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Mirrors > Home > MPE Home > Th. List > sbftALT | Structured version Visualization version GIF version |
Description: Alternate version of sbft 2269. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.ph | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Ref | Expression |
---|---|
sbftALT | ⊢ (Ⅎ𝑥𝜑 → (𝜃 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.ph | . . . 4 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | 1 | spsbeALT 2588 | . . 3 ⊢ (𝜃 → ∃𝑥𝜑) |
3 | 19.9t 2203 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
4 | 2, 3 | syl5ib 246 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜃 → 𝜑)) |
5 | nf5r 2192 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
6 | 1 | stdpc4ALT 2589 | . . 3 ⊢ (∀𝑥𝜑 → 𝜃) |
7 | 5, 6 | syl6 35 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → 𝜃)) |
8 | 4, 7 | impbid 214 | 1 ⊢ (Ⅎ𝑥𝜑 → (𝜃 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → 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-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 |
This theorem is referenced by: sbfALT 2593 |
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