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Mirrors > Home > NFE Home > Th. List > ifclda | GIF version |
Description: Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
ifclda.1 | ⊢ ((φ ∧ ψ) → A ∈ C) |
ifclda.2 | ⊢ ((φ ∧ ¬ ψ) → B ∈ C) |
Ref | Expression |
---|---|
ifclda | ⊢ (φ → if(ψ, A, B) ∈ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3668 | . . . 4 ⊢ (ψ → if(ψ, A, B) = A) | |
2 | 1 | adantl 452 | . . 3 ⊢ ((φ ∧ ψ) → if(ψ, A, B) = A) |
3 | ifclda.1 | . . 3 ⊢ ((φ ∧ ψ) → A ∈ C) | |
4 | 2, 3 | eqeltrd 2427 | . 2 ⊢ ((φ ∧ ψ) → if(ψ, A, B) ∈ C) |
5 | iffalse 3669 | . . . 4 ⊢ (¬ ψ → if(ψ, A, B) = B) | |
6 | 5 | adantl 452 | . . 3 ⊢ ((φ ∧ ¬ ψ) → if(ψ, A, B) = B) |
7 | ifclda.2 | . . 3 ⊢ ((φ ∧ ¬ ψ) → B ∈ C) | |
8 | 6, 7 | eqeltrd 2427 | . 2 ⊢ ((φ ∧ ¬ ψ) → if(ψ, A, B) ∈ C) |
9 | 4, 8 | pm2.61dan 766 | 1 ⊢ (φ → if(ψ, A, B) ∈ C) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 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-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-if 3663 |
This theorem is referenced by: (None) |
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