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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sbccomieg | Structured version Visualization version GIF version | ||
| Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.) |
| Ref | Expression |
|---|---|
| sbccomieg.1 | ⊢ (𝑎 = 𝐴 → 𝐵 = 𝐶) |
| Ref | Expression |
|---|---|
| sbccomieg | ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcex 3798 | . 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 → 𝐴 ∈ V) | |
| 2 | spesbc 3882 | . . 3 ⊢ ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑) | |
| 3 | sbcex 3798 | . . . 4 ⊢ ([𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) | |
| 4 | 3 | exlimiv 1930 | . . 3 ⊢ (∃𝑏[𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) |
| 5 | 2, 4 | syl 17 | . 2 ⊢ ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) |
| 6 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑎𝐶 | |
| 7 | nfsbc1v 3808 | . . . 4 ⊢ Ⅎ𝑎[𝐴 / 𝑎]𝜑 | |
| 8 | 6, 7 | nfsbcw 3810 | . . 3 ⊢ Ⅎ𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑 |
| 9 | sbccomieg.1 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝐵 = 𝐶) | |
| 10 | sbceq1a 3799 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑎]𝜑)) | |
| 11 | 9, 10 | sbceqbid 3795 | . . 3 ⊢ (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑)) |
| 12 | 8, 11 | sbciegf 3827 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑)) |
| 13 | 1, 5, 12 | pm5.21nii 378 | 1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 |
| 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-3an 1089 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-ral 3062 df-rex 3071 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: 2rexfrabdioph 42807 3rexfrabdioph 42808 4rexfrabdioph 42809 6rexfrabdioph 42810 7rexfrabdioph 42811 |
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