Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 1811 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | |
2 | sbequi 1811 | . . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) | |
3 | 2 | equcoms 1684 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) |
4 | 1, 3 | impbid 128 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 [wsb 1735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 |
This theorem is referenced by: drsb2 1813 sbco2vlem 1917 sbco2v 1921 sbco2yz 1936 sbcocom 1943 sb10f 1970 hbsb4 1987 nfsb4or 1998 sb8eu 2012 sb8euh 2022 cbvab 2263 cbvralf 2648 cbvrexf 2649 cbvreu 2652 cbvralsv 2668 cbvrexsv 2669 cbvrab 2684 cbvreucsf 3064 cbvrabcsf 3065 sbss 3471 disjiun 3924 cbvopab1 4001 cbvmpt 4023 tfis 4497 findes 4517 cbviota 5093 sb8iota 5095 cbvriota 5740 uzind4s 9392 bezoutlemmain 11693 cbvrald 13005 setindft 13173 |
Copyright terms: Public domain | W3C validator |