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Mirrors > Home > ILE Home > Th. List > sbequ12 | GIF version |
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbequ12 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ1 1779 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
2 | sbequ2 1780 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑)) | |
3 | 1, 2 | impbid 129 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 |
This theorem depends on definitions: df-bi 117 df-sb 1774 |
This theorem is referenced by: sbequ12r 1783 sbequ12a 1784 sbid 1785 ax16 1824 sb8h 1865 sb8eh 1866 sb8 1867 sb8e 1868 ax16ALT 1870 sbco 1980 sbcomxyyz 1984 sb9v 1990 sb6a 2000 mopick 2116 clelab 2315 sbab 2317 nfabdw 2351 cbvralf 2710 cbvrexf 2711 cbvralsv 2734 cbvrexsv 2735 cbvrab 2750 sbhypf 2801 mob2 2932 reu2 2940 reu6 2941 sbcralt 3054 sbcrext 3055 sbcralg 3056 sbcreug 3058 cbvreucsf 3136 cbvrabcsf 3137 cbvopab1 4091 cbvopab1s 4093 csbopabg 4096 cbvmptf 4112 cbvmpt 4113 opelopabsb 4278 frind 4370 tfis 4600 findes 4620 opeliunxp 4699 ralxpf 4791 rexxpf 4792 cbviota 5201 csbiotag 5228 cbvriota 5861 csbriotag 5863 abrexex2g 6144 opabex3d 6145 opabex3 6146 abrexex2 6148 dfoprab4f 6217 finexdc 6929 ssfirab 6961 uzind4s 9619 zsupcllemstep 11977 bezoutlemmain 12030 nnwosdc 12071 cbvrald 14993 bj-bdfindes 15154 bj-findes 15186 |
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