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Theorem nfunidALT2 34574
Description: Deduction version of nfuni 4474. (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 2799 . . 3 𝑥{𝑦 ∣ ∀𝑥 𝑦𝐴}
21nfuni 4474 . 2 𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴}
3 nfunidALT2.1 . . 3 (𝜑𝑥𝐴)
4 nfnfc1 2796 . . . 4 𝑥𝑥𝐴
5 abidnf 3408 . . . . 5 (𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
65unieqd 4478 . . . 4 (𝑥𝐴 {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
74, 6nfceqdf 2789 . . 3 (𝑥𝐴 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
83, 7syl 17 . 2 (𝜑 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
92, 8mpbii 223 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wcel 2030  {cab 2637  wnfc 2780   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-uni 4469
This theorem is referenced by: (None)
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