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Mirrors > Home > MPE Home > Th. List > sbccow | Structured version Visualization version GIF version |
Description: A composition law for class substitution. Version of sbcco 3798 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 26-Sep-2003.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
sbccow | ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3782 | . 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | sbcex 3782 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
3 | dfsbcq 3774 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝜑)) | |
4 | dfsbcq 3774 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
5 | sbsbc 3776 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
6 | 5 | sbbii 2081 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) |
7 | sbco2vv 2108 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
8 | sbsbc 3776 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) | |
9 | 6, 7, 8 | 3bitr3ri 304 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
10 | sbsbc 3776 | . . . 4 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) | |
11 | 9, 10 | bitri 277 | . . 3 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
12 | 3, 4, 11 | vtoclbg 3569 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
13 | 1, 2, 12 | pm5.21nii 382 | 1 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2069 ∈ wcel 2114 Vcvv 3494 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3773 |
This theorem is referenced by: sbc7 3803 sbccom 3854 sbcralt 3855 csbcow 3898 2nreu 4393 bnj62 31990 bnj610 32018 bnj976 32049 bnj1468 32118 aomclem6 39708 |
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