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Mirrors > Home > ILE Home > Th. List > sbcie | GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
sbcie.1 | ⊢ 𝐴 ∈ V |
sbcie.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbcie | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcie.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbcie.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | sbcieg 2893 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 1, 3 | ax-mp 7 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1299 ∈ wcel 1448 Vcvv 2641 [wsbc 2862 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-sbc 2863 |
This theorem is referenced by: findcard2 6712 findcard2s 6713 ac6sfi 6721 nn1suc 8597 indstr 9238 bezoutlemmain 11479 prmind2 11594 |
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