Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbviinvg | Structured version Visualization version GIF version |
Description: Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2389. Usage of the weaker cbviinv 4959 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 2976 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2976 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbviunvg.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbviing 4957 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∩ ciin 4913 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-iin 4915 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |