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Mirrors > Home > MPE Home > Th. List > cbviin | Structured version Visualization version GIF version |
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2372. See cbviing 5043 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
Ref | Expression |
---|---|
cbviun.1 | ⊢ Ⅎ𝑦𝐵 |
cbviun.2 | ⊢ Ⅎ𝑥𝐶 |
cbviun.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviin | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviun.1 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
2 | 1 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
3 | cbviun.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
5 | cbviun.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
6 | 5 | eleq2d 2820 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
7 | 2, 4, 6 | cbvralw 3304 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
8 | 7 | abbii 2803 | . 2 ⊢ {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
9 | df-iin 5001 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
10 | df-iin 5001 | . 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
11 | 8, 9, 10 | 3eqtr4i 2771 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 ∀wral 3062 ∩ ciin 4999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-iin 5001 |
This theorem is referenced by: cbviinv 5045 elrfirn2 41434 fnlimfvre 44390 smflimlem6 45492 smflim 45493 smflim2 45522 smfsup 45530 smfinflem 45533 smfinf 45534 smflimsup 45544 smfliminf 45547 |
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