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Mirrors > Home > NFE Home > Th. List > cbviin | GIF version |
Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) |
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 2483 | . . . 4 ⊢ Ⅎy z ∈ B |
3 | cbviun.2 | . . . . 5 ⊢ ℲxC | |
4 | 3 | nfcri 2483 | . . . 4 ⊢ Ⅎx z ∈ C |
5 | cbviun.3 | . . . . 5 ⊢ (x = y → B = C) | |
6 | 5 | eleq2d 2420 | . . . 4 ⊢ (x = y → (z ∈ B ↔ z ∈ C)) |
7 | 2, 4, 6 | cbvral 2831 | . . 3 ⊢ (∀x ∈ A z ∈ B ↔ ∀y ∈ A z ∈ C) |
8 | 7 | abbii 2465 | . 2 ⊢ {z ∣ ∀x ∈ A z ∈ B} = {z ∣ ∀y ∈ A z ∈ C} |
9 | df-iin 3972 | . 2 ⊢ ∩x ∈ A B = {z ∣ ∀x ∈ A z ∈ B} | |
10 | df-iin 3972 | . 2 ⊢ ∩y ∈ A C = {z ∣ ∀y ∈ A z ∈ C} | |
11 | 8, 9, 10 | 3eqtr4i 2383 | 1 ⊢ ∩x ∈ A B = ∩y ∈ A C |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 {cab 2339 Ⅎwnfc 2476 ∀wral 2614 ∩ciin 3970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-iin 3972 |
This theorem is referenced by: cbviinv 4006 |
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