Theorem iotabi 4349

Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
iotabi	⊢ (∀x(φ ↔ ψ) → (℩xφ) = (℩xψ))

Proof of Theorem iotabi

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abbi 2464	. . . . . 6 ⊢ (∀x(φ ↔ ψ) ↔ {x ∣ φ} = {x ∣ ψ})
2	1	biimpi 186	. . . . 5 ⊢ (∀x(φ ↔ ψ) → {x ∣ φ} = {x ∣ ψ})
3	2	eqeq1d 2361	. . . 4 ⊢ (∀x(φ ↔ ψ) → ({x ∣ φ} = {z} ↔ {x ∣ ψ} = {z}))
4	3	abbidv 2468	. . 3 ⊢ (∀x(φ ↔ ψ) → {z ∣ {x ∣ φ} = {z}} = {z ∣ {x ∣ ψ} = {z}})
5	4	unieqd 3903	. 2 ⊢ (∀x(φ ↔ ψ) → ∪{z ∣ {x ∣ φ} = {z}} = ∪{z ∣ {x ∣ ψ} = {z}})
6		df-iota 4340	. 2 ⊢ (℩xφ) = ∪{z ∣ {x ∣ φ} = {z}}
7		df-iota 4340	. 2 ⊢ (℩xψ) = ∪{z ∣ {x ∣ ψ} = {z}}
8	5, 6, 7	3eqtr4g 2410	1 ⊢ (∀x(φ ↔ ψ) → (℩xφ) = (℩xψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642  {cab 2339  {csn 3738  ∪cuni 3892  ℩cio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-uni 3893  df-iota 4340

This theorem is referenced by:  iotabidv  4361  iotabii  4362