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| Mirrors > Home > MPE Home > Th. List > sb6x | Structured version Visualization version GIF version | ||
| Description: Equivalence involving substitution for a variable not free. Usage of this theorem is discouraged because it depends on ax-13 2376. Usage of sb6 2084 is preferred, which requires fewer axioms. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| sb6x.1 | ⊢ Ⅎ𝑥𝜑 | 
| Ref | Expression | 
|---|---|
| sb6x | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sb6x.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 1 | sbf 2270 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) | 
| 3 | biidd 262 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
| 4 | 1, 3 | equsal 2421 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) | 
| 5 | 2, 4 | bitr4i 278 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 Ⅎwnf 1782 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-12 2176 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: (None) | 
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