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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 1720 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | |
2 | sbequi 1720 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) | |
3 | 2 | equcoms 1594 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) |
4 | 1, 3 | impbid 120 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ 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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: drsb2 1722 sbco2vlem 1820 sbco2yz 1837 sbcocom 1844 sb10f 1871 hbsb4 1888 nfsb4or 1899 sb8eu 1913 sb8euh 1923 cbvab 2160 cbvralf 2527 cbvrexf 2528 cbvreu 2531 cbvralsv 2544 cbvrexsv 2545 cbvrab 2555 cbvreucsf 2910 cbvrabcsf 2911 sbss 3329 cbvopab1 3830 cbvmpt 3851 tfis 4306 findes 4326 cbviota 4872 sb8iota 4874 cbvriota 5478 uzind4s 8533 cbvrald 9927 setindft 10090 |
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