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Theorem nfunidALT2 33073
Description: Deduction version of nfuni 4368. (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 2751 . . 3 𝑥{𝑦 ∣ ∀𝑥 𝑦𝐴}
21nfuni 4368 . 2 𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴}
3 nfunidALT2.1 . . 3 (𝜑𝑥𝐴)
4 nfnfc1 2749 . . . 4 𝑥𝑥𝐴
5 abidnf 3337 . . . . 5 (𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
65unieqd 4372 . . . 4 (𝑥𝐴 {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
74, 6nfceqdf 2742 . . 3 (𝑥𝐴 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
83, 7syl 17 . 2 (𝜑 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
92, 8mpbii 221 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wcel 1975  {cab 2591  wnfc 2733   cuni 4362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-uni 4363
This theorem is referenced by: (None)
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