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Mirrors > Home > ILE Home > Th. List > sbid | GIF version |
Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbid | ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 1701 | . . 3 ⊢ 𝑥 = 𝑥 | |
2 | sbequ12 1771 | . . 3 ⊢ (𝑥 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑥]𝜑)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝜑 ↔ [𝑥 / 𝑥]𝜑) |
4 | 3 | bicomi 132 | 1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 [wsb 1762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 |
This theorem depends on definitions: df-bi 117 df-sb 1763 |
This theorem is referenced by: abid 2165 sbceq1a 2972 sbcid 2978 |
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