Theorem nfiota1 5182

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 5181	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
2		nfaba1 2325	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
3	2	nfuni 3817	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
4	1, 3	nfcxfr 2316	1 ⊢ Ⅎ𝑥(℩𝑥𝜑)

Syntax hints:   ↔ wb 105  ∀wal 1351  {cab 2163  Ⅎwnfc 2306  ∪ cuni 3811  ℩cio 5178

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-sn 3600  df-uni 3812  df-iota 5180

This theorem is referenced by:  iota2df  5204  sniota  5209  nfriota1  5840  erovlem  6629