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Mirrors > Home > ILE Home > Th. List > iotajust | GIF version |
Description: Soundness justification theorem for df-iota 5083. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
iotajust | ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3533 | . . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤}) | |
2 | 1 | eqeq2d 2149 | . . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤})) |
3 | 2 | cbvabv 2262 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} |
4 | sneq 3533 | . . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧}) | |
5 | 4 | eqeq2d 2149 | . . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧})) |
6 | 5 | cbvabv 2262 | . . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
7 | 3, 6 | eqtri 2158 | . 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
8 | 7 | unieqi 3741 | 1 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 {cab 2123 {csn 3522 ∪ cuni 3731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-sn 3528 df-uni 3732 |
This theorem is referenced by: (None) |
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