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Mirrors > Home > MPE Home > Th. List > nfeld | Structured version Visualization version GIF version |
Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfeqd.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfeqd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfeld | ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clel 2774 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | nfv 1957 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvd 2935 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) | |
4 | nfeqd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeqd 2942 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝐴) |
6 | nfeqd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
7 | 6 | nfcrd 2927 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
8 | 5, 7 | nfand 1944 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 2, 8 | nfexd 2305 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | 1, 9 | nfxfrd 1898 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∃wex 1823 Ⅎwnf 1827 ∈ wcel 2107 Ⅎwnfc 2919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 df-cleq 2770 df-clel 2774 df-nfc 2921 |
This theorem is referenced by: nfel 2946 nfneld 3083 nfrald 3126 ralcom2 3290 nfreud 3298 nfrmod 3299 nfrmo 3301 nfsbc1d 3670 nfsbcd 3673 sbcrext 3729 nfdisj 4866 nfbrd 4932 nfriotad 6891 nfixp 8213 axrepndlem2 9750 axrepnd 9751 axunnd 9753 axpowndlem2 9755 axpowndlem3 9756 axpowndlem4 9757 axpownd 9758 axregndlem2 9760 axinfndlem1 9762 axinfnd 9763 axacndlem4 9767 axacndlem5 9768 axacnd 9769 |
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