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Mirrors > Home > MPE Home > Th. List > iotauni2 | Structured version Visualization version GIF version |
Description: Version of iotauni 6512 using df-iota 6489 instead of dfiota2 6490. (Contributed by SN, 6-Nov-2024.) |
Ref | Expression |
---|---|
iotauni2 | ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iotaval2 6505 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦) | |
2 | unieq 4913 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦}) | |
3 | unisnv 4924 | . . . 4 ⊢ ∪ {𝑦} = 𝑦 | |
4 | 2, 3 | eqtr2di 2783 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑}) |
5 | 1, 4 | eqtrd 2766 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
6 | 5 | exlimiv 1925 | 1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∃wex 1773 {cab 2703 {csn 4623 ∪ cuni 4902 ℩cio 6487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-un 3948 df-in 3950 df-ss 3960 df-sn 4624 df-pr 4626 df-uni 4903 df-iota 6489 |
This theorem is referenced by: iotassuni 6509 |
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