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Mirrors > Home > ILE Home > Th. List > ifeq2 | GIF version |
Description: Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
ifeq2 | ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabeq 2744 | . . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
2 | 1 | uneq2d 3304 | . 2 ⊢ (𝐴 = 𝐵 → ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑})) |
3 | dfif6 3551 | . 2 ⊢ if(𝜑, 𝐶, 𝐴) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) | |
4 | dfif6 3551 | . 2 ⊢ if(𝜑, 𝐶, 𝐵) = ({𝑥 ∈ 𝐶 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ ¬ 𝜑}) | |
5 | 2, 3, 4 | 3eqtr4g 2247 | 1 ⊢ (𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1364 {crab 2472 ∪ cun 3142 ifcif 3549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-un 3148 df-if 3550 |
This theorem is referenced by: ifeq12 3565 ifeq2d 3567 ifbieq2i 3572 xrmaxiflemcom 11288 |
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