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Theorem nfunidALT2 35657
Description: Deduction version of nfuni 4757. (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 2958 . . 3 𝑥{𝑦 ∣ ∀𝑥 𝑦𝐴}
21nfuni 4757 . 2 𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴}
3 nfunidALT2.1 . . 3 (𝜑𝑥𝐴)
4 nfnfc1 2954 . . . 4 𝑥𝑥𝐴
5 abidnf 3635 . . . . 5 (𝑥𝐴 → {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
65unieqd 4761 . . . 4 (𝑥𝐴 {𝑦 ∣ ∀𝑥 𝑦𝐴} = 𝐴)
74, 6nfceqdf 2946 . . 3 (𝑥𝐴 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
83, 7syl 17 . 2 (𝜑 → (𝑥 {𝑦 ∣ ∀𝑥 𝑦𝐴} ↔ 𝑥 𝐴))
92, 8mpbii 234 1 (𝜑𝑥 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1523  wcel 2083  {cab 2777  wnfc 2935   cuni 4751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-uni 4752
This theorem is referenced by: (None)
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