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Mirrors > Home > NFE Home > Th. List > ifeq1 | GIF version |
Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
ifeq1 | ⊢ (A = B → if(φ, A, C) = if(φ, B, C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq 2853 | . . 3 ⊢ (A = B → {x ∈ A ∣ φ} = {x ∈ B ∣ φ}) | |
2 | 1 | uneq1d 3417 | . 2 ⊢ (A = B → ({x ∈ A ∣ φ} ∪ {x ∈ C ∣ ¬ φ}) = ({x ∈ B ∣ φ} ∪ {x ∈ C ∣ ¬ φ})) |
3 | dfif6 3665 | . 2 ⊢ if(φ, A, C) = ({x ∈ A ∣ φ} ∪ {x ∈ C ∣ ¬ φ}) | |
4 | dfif6 3665 | . 2 ⊢ if(φ, B, C) = ({x ∈ B ∣ φ} ∪ {x ∈ C ∣ ¬ φ}) | |
5 | 2, 3, 4 | 3eqtr4g 2410 | 1 ⊢ (A = B → if(φ, A, C) = if(φ, B, C)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1642 {crab 2618 ∪ cun 3207 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: ifeq12 3675 ifeq1d 3676 ifbieq12i 3683 ifexg 3721 |
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