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Mirrors > Home > ILE Home > Th. List > iotajust | GIF version |
Description: Soundness justification theorem for df-iota 5193. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
iotajust | ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3618 | . . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤}) | |
2 | 1 | eqeq2d 2201 | . . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤})) |
3 | 2 | cbvabv 2314 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} |
4 | sneq 3618 | . . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧}) | |
5 | 4 | eqeq2d 2201 | . . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧})) |
6 | 5 | cbvabv 2314 | . . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
7 | 3, 6 | eqtri 2210 | . 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
8 | 7 | unieqi 3834 | 1 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 {cab 2175 {csn 3607 ∪ cuni 3824 |
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-rex 2474 df-sn 3613 df-uni 3825 |
This theorem is referenced by: (None) |
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