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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbccom2f | Structured version Visualization version GIF version | ||
| Description: Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.) | 
| Ref | Expression | 
|---|---|
| sbccom2f.1 | ⊢ 𝐴 ∈ V | 
| sbccom2f.2 | ⊢ Ⅎ𝑦𝐴 | 
| Ref | Expression | 
|---|---|
| sbccom2f | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbccow 3811 | . . . 4 ⊢ ([𝐵 / 𝑧][𝑧 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜑) | |
| 2 | 1 | bicomi 224 | . . 3 ⊢ ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑧][𝑧 / 𝑦]𝜑) | 
| 3 | 2 | sbcbii 3846 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑) | 
| 4 | sbccom2f.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 5 | 4 | sbccom2 38132 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑧][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | 
| 6 | vex 3484 | . . . . . . 7 ⊢ 𝑧 ∈ V | |
| 7 | 6 | sbccom2 38132 | . . . . . 6 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝑧 / 𝑦⦌𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | 
| 8 | sbccom2f.2 | . . . . . . . 8 ⊢ Ⅎ𝑦𝐴 | |
| 9 | 6, 8 | csbgfi 3919 | . . . . . . 7 ⊢ ⦋𝑧 / 𝑦⦌𝐴 = 𝐴 | 
| 10 | dfsbcq 3790 | . . . . . . 7 ⊢ (⦋𝑧 / 𝑦⦌𝐴 = 𝐴 → ([⦋𝑧 / 𝑦⦌𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑)) | |
| 11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ ([⦋𝑧 / 𝑦⦌𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | 
| 12 | 7, 11 | bitri 275 | . . . . 5 ⊢ ([𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥][𝑧 / 𝑦]𝜑) | 
| 13 | 12 | bicomi 224 | . . . 4 ⊢ ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | 
| 14 | 13 | sbcbii 3846 | . . 3 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑) | 
| 15 | sbccow 3811 | . . 3 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝑧 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | |
| 16 | 14, 15 | bitri 275 | . 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑧][𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | 
| 17 | 3, 5, 16 | 3bitri 297 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 [wsbc 3788 ⦋csb 3899 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 | 
| This theorem is referenced by: sbccom2fi 38134 | 
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