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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 1717 | . 2 ⊢ (x = y → ([x / z]φ → [y / z]φ)) | |
2 | sbequi 1717 | . . 3 ⊢ (y = x → ([y / z]φ → [x / z]φ)) | |
3 | 2 | equcoms 1591 | . 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 1642 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 |
This theorem is referenced by: drsb2 1719 sbco2vlem 1817 sbco2yz 1834 sbcocom 1841 sb10f 1868 hbsb4 1885 nfsb4or 1896 sb8eu 1910 sb8euh 1920 cbvab 2157 cbvralf 2521 cbvrexf 2522 cbvreu 2525 cbvralsv 2538 cbvrexsv 2539 cbvrab 2549 cbvreucsf 2904 cbvrabcsf 2905 sbss 3323 cbvopab1 3821 cbvmpt 3842 tfis 4249 findes 4269 cbviota 4815 sb8iota 4817 cbvriota 5421 uzind4s 8309 cbvrald 9262 setindft 9425 |
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