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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 35406 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) |
4 | sbccom2fi.3 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 | |
5 | dfsbcq 3776 | . . 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 3831 | . 2 ⊢ ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦]𝜓) |
9 | 3, 6, 8 | 3bitri 299 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 Vcvv 3496 [wsbc 3774 ⦋csb 3885 |
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-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-csb 3886 |
This theorem is referenced by: csbcom2fi 35408 |
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