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Mirrors > Home > NFE Home > Th. List > iineq2i | GIF version |
Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
iuneq2i.1 | ⊢ (x ∈ A → B = C) |
Ref | Expression |
---|---|
iineq2i | ⊢ ∩x ∈ A B = ∩x ∈ A C |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iineq2 3986 | . 2 ⊢ (∀x ∈ A B = C → ∩x ∈ A B = ∩x ∈ A C) | |
2 | iuneq2i.1 | . 2 ⊢ (x ∈ A → B = C) | |
3 | 1, 2 | mprg 2683 | 1 ⊢ ∩x ∈ A B = ∩x ∈ A C |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 ∩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-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-ral 2619 df-iin 3972 |
This theorem is referenced by: iinrab 4028 iinin1 4037 |
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