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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 3730 | . 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 → 𝐴 ∈ V) | |
2 | spesbc 3811 | . . 3 ⊢ ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑) | |
3 | sbcex 3730 | . . . 4 ⊢ ([𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) | |
4 | 3 | exlimiv 1931 | . . 3 ⊢ (∃𝑏[𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) |
5 | 2, 4 | syl 17 | . 2 ⊢ ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → 𝐴 ∈ V) |
6 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑎𝐶 | |
7 | nfsbc1v 3740 | . . . 4 ⊢ Ⅎ𝑎[𝐴 / 𝑎]𝜑 | |
8 | 6, 7 | nfsbcw 3742 | . . 3 ⊢ Ⅎ𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑 |
9 | sbccomieg.1 | . . . 4 ⊢ (𝑎 = 𝐴 → 𝐵 = 𝐶) | |
10 | sbceq1a 3731 | . . . 4 ⊢ (𝑎 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑎]𝜑)) | |
11 | 9, 10 | sbceqbid 3727 | . . 3 ⊢ (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑)) |
12 | 8, 11 | sbciegf 3757 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑)) |
13 | 1, 5, 12 | pm5.21nii 383 | 1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑 ↔ [𝐶 / 𝑏][𝐴 / 𝑎]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 |
This theorem is referenced by: 2rexfrabdioph 39737 3rexfrabdioph 39738 4rexfrabdioph 39739 6rexfrabdioph 39740 7rexfrabdioph 39741 |
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