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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 5167 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfe1 2147 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
3 | 2 | nfab 2912 | . 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
4 | 1, 3 | nfcxfr 2904 | 1 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∃wex 1781 {cab 2713 Ⅎwnfc 2886 〈cop 4591 {copab 5166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-opab 5167 |
This theorem is referenced by: nfmpt1 5212 rexopabb 5484 ssopab2bw 5503 ssopab2b 5505 0nelopabOLD 5524 dmopab 5870 rnopab 5908 funopab 6534 fvopab5 6978 zfrep6 7884 opabdm 31428 opabrn 31429 fpwrelmap 31545 fineqvrep 33587 bj-opabco 35648 vvdifopab 36709 aomclem8 41364 sprsymrelf 45657 |
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