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Mirrors > Home > ILE Home > Th. List > sbcco | GIF version |
Description: A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbcco | ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2912 | . 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | sbcex 2912 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
3 | dfsbcq 2906 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝜑)) | |
4 | dfsbcq 2906 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
5 | sbsbc 2908 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
6 | 5 | sbbii 1738 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) |
7 | nfv 1508 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
8 | 7 | sbco2 1936 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
9 | sbsbc 2908 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) | |
10 | 6, 8, 9 | 3bitr3ri 210 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
11 | sbsbc 2908 | . . . 4 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
12 | 10, 11 | bitri 183 | . . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
13 | 3, 4, 12 | vtoclbg 2742 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
14 | 1, 2, 13 | pm5.21nii 693 | 1 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 [wsb 1735 Vcvv 2681 [wsbc 2904 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-sbc 2905 |
This theorem is referenced by: sbc7 2930 sbccom 2979 sbcralt 2980 sbcrext 2981 csbco 3008 |
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