Theorem riota1a 7255

Description: Property of iota. (Contributed by NM, 23-Aug-2011.)

Assertion

Ref	Expression
riota1a	⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))

Proof of Theorem riota1a

Step	Hyp	Ref	Expression
1		ibar 529	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2		df-reu 3072	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		iota1 6410	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
4	2, 3	sylbi 216	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))
5	1, 4	sylan9bb 510	1 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (𝜑 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃!weu 2568  ∃!wreu 3066  ℩cio 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-reu 3072  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840  df-iota 6391

This theorem is referenced by: (None)