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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 2978 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 [wsbc 2946 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-sbc 2947 |
This theorem is referenced by: findcard2 6846 findcard2s 6847 ac6sfi 6855 nn1suc 8867 indstr 9522 bezoutlemmain 11916 prmind2 12031 |
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