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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 2390. Check out sbid2vw 2260 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 2549 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
2 | sbid2.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | sbf 2271 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
4 | 1, 3 | bitri 277 | 1 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 Ⅎwnf 1784 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: sbid2v 2551 sbtrt 2557 |
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