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| Mirrors > Home > MPE Home > Th. List > sb4av | Structured version Visualization version GIF version | ||
| Description: Version of sb4a 2485 with a disjoint variable condition, which does not require ax-13 2377. The distinctor antecedent from sb4b 2480 is replaced by a disjoint variable condition in this theorem. (Contributed by NM, 2-Feb-2007.) (Revised by BJ, 15-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| sb4av | ⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sp 2183 | . . 3 ⊢ (∀𝑡𝜑 → 𝜑) | |
| 2 | 1 | sbimi 2074 | . 2 ⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → [𝑡 / 𝑥]𝜑) | 
| 3 | sb6 2085 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) | |
| 4 | 2, 3 | sylib 218 | 1 ⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 [wsb 2064 | 
| 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 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 | 
| This theorem is referenced by: bj-hbsb2av 36815 | 
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