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Theorem nfunidALT2 34778
Description: Deduction version of nfuni 4580. (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 2919 . . 3 𝑥{𝑦 ∣ ∀𝑥 𝑦𝐴}
21nfuni 4580 . 2 𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴}
3 nfunidALT2.1 . . 3 (𝜑𝑥𝐴)
4 nfnfc1 2916 . . . 4 𝑥𝑥𝐴
5 abidnf 3527 . . . . 5 (𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
65unieqd 4584 . . . 4 (𝑥𝐴 {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
74, 6nfceqdf 2909 . . 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 1629  wcel 2145  {cab 2757  wnfc 2900   cuni 4574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-uni 4575
This theorem is referenced by: (None)
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