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Mirrors > Home > ILE Home > Th. List > iotajust | GIF version |
Description: Soundness justification theorem for df-iota 5160. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
iotajust | ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3594 | . . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤}) | |
2 | 1 | eqeq2d 2182 | . . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤})) |
3 | 2 | cbvabv 2295 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} |
4 | sneq 3594 | . . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧}) | |
5 | 4 | eqeq2d 2182 | . . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧})) |
6 | 5 | cbvabv 2295 | . . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
7 | 3, 6 | eqtri 2191 | . 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
8 | 7 | unieqi 3806 | 1 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 {cab 2156 {csn 3583 ∪ cuni 3796 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-sn 3589 df-uni 3797 |
This theorem is referenced by: (None) |
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