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Mirrors > Home > ILE Home > Th. List > sbc2ie | GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
sbc2ie.1 | ⊢ 𝐴 ∈ V |
sbc2ie.2 | ⊢ 𝐵 ∈ V |
sbc2ie.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbc2ie | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc2ie.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbc2ie.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | nfv 1515 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1515 | . . 3 ⊢ Ⅎ𝑦𝜓 | |
5 | 2 | nfth 1451 | . . 3 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | sbc2ie.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
7 | 3, 4, 5, 6 | sbc2iegf 3016 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
8 | 1, 2, 7 | mp2an 423 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ 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: sbc3ie 3019 |
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