Theorem iota1 6516

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 2600	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		sp 2220	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = 𝑧))
3		iotaval 6511	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (℩𝑥𝜑) = 𝑧)
4	3	eqeq2d 2772	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝑥 = (℩𝑥𝜑) ↔ 𝑥 = 𝑧))
5	2, 4	bitr4d 285	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ 𝑥 = (℩𝑥𝜑)))
6		eqcom 2768	. . . 4 ⊢ (𝑥 = (℩𝑥𝜑) ↔ (℩𝑥𝜑) = 𝑥)
7	5, 6	bitrdi 290	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
8	7	exlimiv 1963	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))
9	1, 8	sylbi 220	1 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812  ∃!weu 2594  ℩cio 6491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-uni 4868  df-iota 6493

This theorem is used by:  iota2df  6524  sniota  6528  tz6.12c  6905  opabiota  6965  riota1  7396  riota1a  7397  erovlem  8827  gsumval3lem2  20113  bnj1366  35452  funressndmafv2rn  48262  tz6.12-afv2  48279