Theorem iota2d 5245

Description: A condition that allows us to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
iota2df.2	⊢ (𝜑 → ∃!𝑥𝜓)
iota2df.3	⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
iota2d	⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem iota2d

Step	Hyp	Ref	Expression
1		iota2df.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
2		iota2df.2	. 2 ⊢ (𝜑 → ∃!𝑥𝜓)
3		iota2df.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
4		nfv 1542	. 2 ⊢ Ⅎ𝑥𝜑
5		nfvd 1543	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6		nfcvd 2340	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
7	1, 2, 3, 4, 5, 6	iota2df 5244	1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃!weu 2045   ∈ wcel 2167  ℩cio 5217

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219

This theorem is referenced by: (None)