Theorem nfiota1 4889

Description: Bound-variable hypothesis builder for the ℩ class. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Assertion

Ref	Expression
nfiota1	⊢ Ⅎ𝑥(℩𝑥𝜑)

Proof of Theorem nfiota1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 4888	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
2		nfaba1 2224	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
3	2	nfuni 3607	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
4	1, 3	nfcxfr 2216	1 ⊢ Ⅎ𝑥(℩𝑥𝜑)

Syntax hints:   ↔ wb 103  ∀wal 1282  {cab 2067  Ⅎwnfc 2206  ∪ cuni 3601  ℩cio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-sn 3404  df-uni 3602  df-iota 4887

This theorem is referenced by:  iota2df  4911  sniota  4914  nfriota1  5495  erovlem  6221