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Mirrors > Home > MPE Home > Th. List > nfopab2 | Structured version Visualization version GIF version |
Description: The second 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 |
---|---|
nfopab2 | ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-opab 5093 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfe1 2151 | . . . 4 ⊢ Ⅎ𝑦∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
3 | 2 | nfex 2332 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfab 2961 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
5 | 1, 4 | nfcxfr 2953 | 1 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 {cab 2776 Ⅎwnfc 2936 〈cop 4531 {copab 5092 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-opab 5093 |
This theorem is referenced by: rexopabb 5380 opelopabsb 5382 ssopab2bw 5399 ssopab2b 5401 0nelopab 5417 dmopab 5748 rnopab 5790 funopab 6359 fvopab5 6777 zfrep6 7638 opabdm 30375 opabrn 30376 fpwrelmap 30495 bj-opabco 34603 vvdifopab 35681 aomclem8 40005 areaquad 40166 sprsymrelf 44012 |
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