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Mirrors > Home > MPE Home > Th. List > sbid2 | Structured version Visualization version GIF version |
Description: An identity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2367. Check out sbid2vw 2246 for a weaker version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbid2.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
sbid2 | ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco 2502 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
2 | sbid2.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | sbf 2258 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 1, 3 | bitri 275 | 1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: sbid2v 2504 sbtrt 2510 |
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