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| Mirrors > Home > MPE Home > Th. List > Mathboxes > euabsn2w | Structured version Visualization version GIF version | ||
| Description: Replace ax-10 2141, ax-11 2157, ax-12 2177 in euabsn2 4725 with substitution hypotheses. (Contributed by SN, 27-May-2025.) |
| Ref | Expression |
|---|---|
| absnw.y | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| euabsn2w.z | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) |
| Ref | Expression |
|---|---|
| euabsn2w | ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | euabsn2w.z | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) | |
| 2 | absnw.y | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 3 | 1, 2 | eu6w 42686 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
| 4 | 1 | absnw 42688 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
| 5 | 4 | exbii 1848 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
| 6 | 3, 5 | bitr4i 278 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 = wceq 1540 ∃wex 1779 ∃!weu 2568 {cab 2714 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sn 4627 |
| This theorem is referenced by: sn-tz6.12-2 42690 |
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