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Mirrors > Home > MPE Home > Th. List > sbalexOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbalex 2239 as of 14-Aug-2025. (Contributed by NM, 14-Apr-2008.) (Revised by BJ, 20-Dec-2020.) (Revised by BJ, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbalexOLD | ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2148 | . . 3 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) | |
2 | ax12v2 2176 | . . . 4 ⊢ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | |
3 | 2 | imp 406 | . . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
4 | 1, 3 | exlimi 2214 | . 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
5 | equs4v 1996 | . 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑)) | |
6 | 4, 5 | impbii 209 | 1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1534 ∃wex 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-10 2138 ax-12 2174 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1776 df-nf 1780 |
This theorem is referenced by: (None) |
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