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Mirrors > Home > MPE Home > Th. List > iotajust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-iota 6316. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
iotajust | ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4579 | . . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤}) | |
2 | 1 | eqeq2d 2834 | . . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤})) |
3 | 2 | cbvabv 2891 | . . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} |
4 | sneq 4579 | . . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧}) | |
5 | 4 | eqeq2d 2834 | . . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧})) |
6 | 5 | cbvabv 2891 | . . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
7 | 3, 6 | eqtri 2846 | . 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
8 | 7 | unieqi 4853 | 1 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {cab 2801 {csn 4569 ∪ cuni 4840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-ss 3954 df-sn 4570 df-uni 4841 |
This theorem is referenced by: (None) |
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