Theorem riota1 6031

Description: Property of restricted iota. Compare iota1 5332. (Contributed by Mario Carneiro, 15-Oct-2016.)

Assertion

Ref	Expression
riota1	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riota1

Step	Hyp	Ref	Expression
1		df-reu 2529	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		iota1 5332	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
3	1, 2	sylbi 121	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
4		df-riota 6011	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	4	eqeq1i 2242	. 2 ⊢ ((℩𝑥 ∈ 𝐴 𝜑) = 𝑥 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥)
6	3, 5	bitr4di 198	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃!weu 2082   ∈ wcel 2205  ∃!wreu 2524  ℩cio 5315  ℩crio 6010

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-reu 2529  df-v 2817  df-sbc 3046  df-un 3218  df-sn 3700  df-pr 3701  df-uni 3920  df-iota 5317  df-riota 6011

This theorem is referenced by:  supelti  7306  oddpwdclemdvds  12892  oddpwdclemndvds  12893