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| Mirrors > Home > MPE Home > Th. List > nfopab1 | Structured version Visualization version GIF version | ||
| Description: The first abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) |
| Ref | Expression |
|---|---|
| nfopab1 | ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-opab 5206 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 2 | nfe1 2150 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
| 3 | 2 | nfab 2911 | . 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
| 4 | 1, 3 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 {cab 2714 Ⅎwnfc 2890 〈cop 4632 {copab 5205 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-opab 5206 |
| This theorem is referenced by: nfmpt1 5250 rexopabb 5533 ssopab2bw 5552 ssopab2b 5554 dmopab 5926 rnopab 5965 funopab 6601 fvopab5 7049 zfrep6 7979 opabdm 32623 opabrn 32624 fpwrelmap 32744 fineqvrep 35109 bj-opabco 37189 vvdifopab 38261 aomclem8 43073 sprsymrelf 47482 |
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