Theorem riota2f 5923

Description: This theorem shows a condition that allows us to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota2f.1	⊢ Ⅎ𝑥𝐵
riota2f.2	⊢ Ⅎ𝑥𝜓
riota2f.3	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
riota2f	⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem riota2f

Step	Hyp	Ref	Expression
1		riota2f.1	. . 3 ⊢ Ⅎ𝑥𝐵
2	1	nfel1 2359	. 2 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴
3	1	a1i 9	. 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝐵)
4		riota2f.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	4	a1i 9	. 2 ⊢ (𝐵 ∈ 𝐴 → Ⅎ𝑥𝜓)
6		id 19	. 2 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴)
7		riota2f.3	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
8	7	adantl 277	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵) → (𝜑 ↔ 𝜓))
9	2, 3, 5, 6, 8	riota2df 5922	1 ⊢ ((𝐵 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜓 ↔ (℩𝑥 ∈ 𝐴 𝜑) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  Ⅎwnf 1483   ∈ wcel 2176  Ⅎwnfc 2335  ∃!wreu 2486  ℩crio 5900

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-reu 2491  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5233  df-riota 5901

This theorem is referenced by:  riota2  5924  riotaprop  5925  riotass2  5928  riotass  5929