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Mirrors > Home > MPE Home > Th. List > sbcid | Structured version Visualization version GIF version |
Description: An identity theorem for substitution. See sbid 2248. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Ref | Expression |
---|---|
sbcid | ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbsbc 3748 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑) | |
2 | sbid 2248 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
3 | 1, 2 | bitr3i 277 | 1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2068 [wsbc 3744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-sbc 3745 |
This theorem is referenced by: csbid 3873 snfil 23231 ex-natded9.26 29405 bnj605 33559 dedths 37453 frege93 42302 or2expropbilem1 45340 |
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