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Mirrors > Home > ILE Home > Th. List > sbequ | GIF version |
Description: An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbequ | ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequi 1760 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | |
2 | sbequi 1760 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) | |
3 | 2 | equcoms 1634 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) |
4 | 1, 3 | impbid 127 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: drsb2 1762 sbco2vlem 1861 sbco2yz 1878 sbcocom 1885 sb10f 1912 hbsb4 1929 nfsb4or 1940 sb8eu 1954 sb8euh 1964 cbvab 2201 cbvralf 2571 cbvrexf 2572 cbvreu 2575 cbvralsv 2588 cbvrexsv 2589 cbvrab 2599 cbvreucsf 2966 cbvrabcsf 2967 sbss 3349 cbvopab1 3851 cbvmpt 3872 tfis 4324 findes 4344 cbviota 4892 sb8iota 4894 cbvriota 5498 uzind4s 8678 bezoutlemmain 10387 cbvrald 10598 setindft 10760 |
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