Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2618 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | nfv 1843 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvd 2765 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) | |
4 | nfeqd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeqd 2772 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝐴) |
6 | nfeqd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
7 | 6 | nfcrd 2771 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
8 | 5, 7 | nfand 1826 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 2, 8 | nfexd 2167 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | 1, 9 | nfxfrd 1780 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-cleq 2615 df-clel 2618 df-nfc 2753 |
This theorem is referenced by: nfel 2777 nfneld 2905 nfrald 2944 ralcom2 3104 nfreud 3112 nfrmod 3113 nfrmo 3115 nfsbc1d 3453 nfsbcd 3456 sbcrext 3511 sbcrextOLD 3512 nfdisj 4632 nfbrd 4698 nfriotad 6619 nfixp 7927 axrepndlem2 9415 axrepnd 9416 axunnd 9418 axpowndlem2 9420 axpowndlem3 9421 axpowndlem4 9422 axpownd 9423 axregndlem2 9425 axinfndlem1 9427 axinfnd 9428 axacndlem4 9432 axacndlem5 9433 axacnd 9434 |
Copyright terms: Public domain | W3C validator |