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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 5210 | . 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 2907 | . 2 ⊢ Ⅎ𝑥{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} |
4 | 1, 3 | nfcxfr 2899 | 1 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1539 ∃wex 1779 {cab 2707 Ⅎwnfc 2881 ⟨cop 4633 {copab 5209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-opab 5210 |
This theorem is referenced by: nfmpt1 5255 rexopabb 5527 ssopab2bw 5546 ssopab2b 5548 0nelopabOLD 5567 dmopab 5914 rnopab 5952 funopab 6582 fvopab5 7029 zfrep6 7943 opabdm 32107 opabrn 32108 fpwrelmap 32225 fineqvrep 34393 bj-opabco 36372 vvdifopab 37431 aomclem8 42105 sprsymrelf 46461 |
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