Theorem nfiota1 5256

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 5255	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
2		nfaba1 2358	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
3	2	nfuni 3873	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
4	1, 3	nfcxfr 2349	1 ⊢ Ⅎ𝑥(℩𝑥𝜑)

Syntax hints:   ↔ wb 105  ∀wal 1373  {cab 2195  Ⅎwnfc 2339  ∪ cuni 3867  ℩cio 5252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-rex 2494  df-sn 3652  df-uni 3868  df-iota 5254

This theorem is referenced by:  iota2df  5280  sniota  5285  nfriota1  5935  erovlem  6744