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Mirrors > Home > NFE Home > Th. List > ifexg | GIF version |
Description: Conditional operator existence. (Contributed by NM, 21-Mar-2011.) |
Ref | Expression |
---|---|
ifexg | ⊢ ((A ∈ V ∧ B ∈ W) → if(φ, A, B) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1 3666 | . . 3 ⊢ (x = A → if(φ, x, y) = if(φ, A, y)) | |
2 | 1 | eleq1d 2419 | . 2 ⊢ (x = A → ( if(φ, x, y) ∈ V ↔ if(φ, A, y) ∈ V)) |
3 | ifeq2 3667 | . . 3 ⊢ (y = B → if(φ, A, y) = if(φ, A, B)) | |
4 | 3 | eleq1d 2419 | . 2 ⊢ (y = B → ( if(φ, A, y) ∈ V ↔ if(φ, A, B) ∈ V)) |
5 | vex 2862 | . . 3 ⊢ x ∈ V | |
6 | vex 2862 | . . 3 ⊢ y ∈ V | |
7 | 5, 6 | ifex 3720 | . 2 ⊢ if(φ, x, y) ∈ V |
8 | 2, 4, 7 | vtocl2g 2918 | 1 ⊢ ((A ∈ V ∧ B ∈ W) → if(φ, A, B) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ifcif 3662 |
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-nan 1288 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-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-if 3663 |
This theorem is referenced by: (None) |
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