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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 5166 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfe1 2147 | . . . 4 ⊢ Ⅎ𝑦∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
3 | 2 | nfex 2318 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfab 2911 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
5 | 1, 4 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∃wex 1781 {cab 2714 Ⅎwnfc 2885 〈cop 4590 {copab 5165 |
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 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-opab 5166 |
This theorem is referenced by: rexopabb 5483 ssopab2bw 5502 ssopab2b 5504 0nelopabOLD 5523 dmopab 5869 rnopab 5907 funopab 6533 fvopab5 6977 zfrep6 7879 opabdm 31359 opabrn 31360 fpwrelmap 31476 fineqvrep 33508 bj-opabco 35597 vvdifopab 36658 aomclem8 41297 areaquad 41459 sprsymrelf 45588 |
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