Theorem iota5 6523

Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.)

Hypothesis

Ref	Expression
iota5.1	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))

Assertion

Ref	Expression
iota5	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem iota5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iota5.1	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))
2	1	alrimiv 1959	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑥(𝜓 ↔ 𝑥 = 𝐴))
3		eqeq2 2772	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
4	3	bibi2d 345	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜓 ↔ 𝑥 = 𝐴)))
5	4	albidv 1952	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝐴)))
6		eqeq2 2772	. . . . 5 ⊢ (𝑦 = 𝐴 → ((℩𝑥𝜓) = 𝑦 ↔ (℩𝑥𝜓) = 𝐴))
7	5, 6	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦) ↔ (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴)))
8		iotaval 6514	. . . 4 ⊢ (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) → (℩𝑥𝜓) = 𝑦)
9	7, 8	vtoclg 3517	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
10	9	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∀𝑥(𝜓 ↔ 𝑥 = 𝐴) → (℩𝑥𝜓) = 𝐴))
11	2, 10	mpd 16	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ℩cio 6494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-un 4150  df-sn 4585  df-pr 4587  df-uni 4867  df-iota 6496

This theorem is used by:  isf32lem9  10444  rlimdm  15725  fsum  15893  fprod  16115  gsumval2a  18881  dchrptlem1  27598  lgsdchrval  27688  iota0def  48215  rlimdmafv  48354  rlimdmafv2  48435