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Theorem nfunidALT2 38496
Description: Deduction version of nfuni 4910. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nfunidALT2.1 (𝜑𝑥𝐴)
Assertion
Ref Expression
nfunidALT2 (𝜑𝑥 𝐴)

Proof of Theorem nfunidALT2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2900 . . 3 𝑥{𝑦 ∣ ∀𝑥 𝑦𝐴}
21nfuni 4910 . 2 𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴}
3 nfunidALT2.1 . . 3 (𝜑𝑥𝐴)
4 nfnfc1 2895 . . . 4 𝑥𝑥𝐴
5 abidnf 3690 . . . . 5 (𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
65unieqd 4916 . . . 4 (𝑥𝐴 {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
74, 6nfceqdf 2887 . . 3 (𝑥𝐴 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
83, 7syl 17 . 2 (𝜑 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
92, 8mpbii 232 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wcel 2098  {cab 2702  wnfc 2875   cuni 4903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-v 3465  df-ss 3957  df-uni 4904
This theorem is referenced by: (None)
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