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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 5116 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfe1 2151 | . . 3 ⊢ Ⅎ𝑥∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
3 | 2 | nfab 2910 | . 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
4 | 1, 3 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∃wex 1787 {cab 2714 Ⅎwnfc 2884 〈cop 4547 {copab 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-opab 5116 |
This theorem is referenced by: nfmpt1 5153 rexopabb 5409 ssopab2bw 5428 ssopab2b 5430 0nelopabOLD 5447 dmopab 5784 rnopab 5823 funopab 6415 fvopab5 6850 zfrep6 7728 opabdm 30670 opabrn 30671 fpwrelmap 30788 fineqvrep 32777 bj-opabco 35094 vvdifopab 36136 aomclem8 40589 sprsymrelf 44620 |
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