|   | Mathbox for Giovanni Mascellani | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbccom2fi | Structured version Visualization version GIF version | ||
| Description: Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| sbccom2fi.1 | ⊢ 𝐴 ∈ V | 
| sbccom2fi.2 | ⊢ Ⅎ𝑦𝐴 | 
| sbccom2fi.3 | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 | 
| sbccom2fi.4 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | 
| Ref | Expression | 
|---|---|
| sbccom2fi | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbccom2fi.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 2 | sbccom2fi.2 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 3 | 1, 2 | sbccom2f 38133 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | 
| 4 | sbccom2fi.3 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 | |
| 5 | dfsbcq 3790 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) | |
| 6 | 4, 5 | ax-mp 5 | . 2 ⊢ ([⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑) | 
| 7 | sbccom2fi.4 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) | |
| 8 | 7 | sbcbii 3846 | . 2 ⊢ ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦]𝜓) | 
| 9 | 3, 6, 8 | 3bitri 297 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓) | 
| 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: csbcom2fi 38135 | 
| Copyright terms: Public domain | W3C validator |