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| Description: Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2377. Usage of the weaker cbviinv 5041 is preferred. (Contributed by Jeff Hankins, 26-Aug-2009.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbviunvg.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| cbviinvg | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcv 2905 | . 2 ⊢ Ⅎ𝑦𝐵 | |
| 2 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝐶 | |
| 3 | cbviunvg.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 4 | 1, 2, 3 | cbviing 5039 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∩ ciin 4992 | 
| 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-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-iin 4994 | 
| This theorem is referenced by: (None) | 
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