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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 (class builder) 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 5120 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfe1 2145 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
3 | 2 | nfab 2981 | . 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
4 | 1, 3 | nfcxfr 2972 | 1 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∃wex 1771 {cab 2796 Ⅎwnfc 2958 〈cop 4563 {copab 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-opab 5120 |
This theorem is referenced by: nfmpt1 5155 rexopabb 5406 opelopabsb 5408 ssopab2bw 5425 ssopab2b 5427 0nelopab 5443 dmopab 5777 rnopab 5819 funopab 6383 fvopab5 6792 zfrep6 7645 opabdm 30290 opabrn 30291 fpwrelmap 30395 vvdifopab 35402 aomclem8 39539 sprsymrelf 43534 |
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