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Mathbox for Stefan O'Rear |
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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 3786 | . 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 → 𝐴 ∈ V) | |
2 | spesbc 3875 | . . 3 ⊢ ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑) | |
3 | sbcex 3786 | . . . 4 ⊢ ([𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) | |
4 | 3 | exlimiv 1926 | . . 3 ⊢ (∃𝑏[𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) |
5 | 2, 4 | syl 17 | . 2 ⊢ ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) |
6 | nfcv 2892 | . . . 4 ⊢ Ⅎ𝑎𝐶 | |
7 | nfsbc1v 3796 | . . . 4 ⊢ Ⅎ𝑎[𝐴 / 𝑎]𝜑 | |
8 | 6, 7 | nfsbcw 3798 | . . 3 ⊢ Ⅎ𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑 |
9 | sbccomieg.1 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝐵 = 𝐶) | |
10 | sbceq1a 3787 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑎]𝜑)) | |
11 | 9, 10 | sbceqbid 3783 | . . 3 ⊢ (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑)) |
12 | 8, 11 | sbciegf 3816 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑)) |
13 | 1, 5, 12 | pm5.21nii 377 | 1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3462 [wsbc 3776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-v 3464 df-sbc 3777 |
This theorem is referenced by: 2rexfrabdioph 42453 3rexfrabdioph 42454 4rexfrabdioph 42455 6rexfrabdioph 42456 7rexfrabdioph 42457 |
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