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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 1589 | . . 3 ⊢ 𝑥 = 𝑥 | |
2 | sbequ12 1654 | . . 3 ⊢ (𝑥 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑥]𝜑)) | |
3 | 1, 2 | ax-mp 7 | . 2 ⊢ (𝜑 ↔ [𝑥 / 𝑥]𝜑) |
4 | 3 | bicomi 123 | 1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 |
This theorem depends on definitions: df-bi 110 df-sb 1646 |
This theorem is referenced by: abid 2028 sbceq1a 2773 sbcid 2779 |
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