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Mirrors > Home > ILE Home > Th. List > sbc2iegf | GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
sbc2iegf.1 | ⊢ Ⅎ𝑥𝜓 |
sbc2iegf.2 | ⊢ Ⅎ𝑦𝜓 |
sbc2iegf.3 | ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊 |
sbc2iegf.4 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbc2iegf | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
2 | simpl 109 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊) | |
3 | sbc2iegf.4 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
4 | 3 | adantll 476 | . . . 4 ⊢ (((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
5 | nfv 1528 | . . . 4 ⊢ Ⅎ𝑦(𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) | |
6 | sbc2iegf.2 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
7 | 6 | a1i 9 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝜓) |
8 | 2, 4, 5, 7 | sbciedf 2998 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
9 | 8 | adantll 476 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
10 | nfv 1528 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉 | |
11 | sbc2iegf.3 | . . 3 ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊 | |
12 | 10, 11 | nfan 1565 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) |
13 | sbc2iegf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
14 | 13 | a1i 9 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Ⅎ𝑥𝜓) |
15 | 1, 9, 12, 14 | sbciedf 2998 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 Ⅎwnf 1460 ∈ wcel 2148 [wsbc 2962 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-sbc 2963 |
This theorem is referenced by: sbc2ie 3034 opelopabaf 4272 |
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