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 non free 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 33101 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) |
4 | sbccom2fi.3 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 | |
5 | dfsbcq 3404 | . . 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 3458 | . 2 ⊢ ([𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ↔ [𝐶 / 𝑦]𝜓) |
9 | 3, 6, 8 | 3bitri 285 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 [wsbc 3402 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 |
This theorem is referenced by: csbcom2fi 33104 |
Copyright terms: Public domain | W3C validator |