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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 (class builder) 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 4644 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
2 | nfe1 2014 | . . . 4 ⊢ Ⅎ𝑦∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) | |
3 | 2 | nfex 2140 | . . 3 ⊢ Ⅎ𝑦∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) |
4 | 3 | nfab 2755 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
5 | 1, 4 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∃wex 1695 {cab 2596 Ⅎwnfc 2738 〈cop 4131 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-opab 4644 |
This theorem is referenced by: opelopabsb 4910 ssopab2b 4927 dmopab 5257 rnopab 5291 funopab 5837 fvopab5 6217 0neqopab 6596 zfrep6 7027 opabdm 28803 opabrn 28804 fpwrelmap 28896 aomclem8 36649 areaquad 36821 |
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