Theorem iota1 6471

Description: Property of iota. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
iota1	⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

Proof of Theorem iota1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2575	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		sp 2191	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = 𝑧))
3		iotaval 6466	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
4	3	eqeq2d 2748	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧))
5	2, 4	bitr4d 282	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
6		eqcom 2744	. . . 4 ⊢ (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥)
7	5, 6	bitrdi 287	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
8	7	exlimiv 1932	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
9	1, 8	sylbi 217	1 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  ∃wex 1781  ∃!weu 2569  ℩cio 6446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-ss 3907  df-sn 4569  df-pr 4571  df-uni 4852  df-iota 6448

This theorem is referenced by:  iota2df  6479  sniota  6483  tz6.12c  6856  opabiota  6916  riota1  7338  riota1a  7339  erovlem  8753  gsumval3lem2  19872  bnj1366  34987  funressndmafv2rn  47683  tz6.12-afv2  47700