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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 2220. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Ref | Expression |
---|---|
sbcid | ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbsbc 3710 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑥]𝜑) | |
2 | sbid 2220 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
3 | 1, 2 | bitr3i 278 | 1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 [wsb 2042 [wsbc 3706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-sbc 3707 |
This theorem is referenced by: csbid 3823 snfil 22156 ex-natded9.26 27890 bnj605 31795 dedths 35629 frege93 39787 or2expropbilem1 42783 |
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