Theorem iotaeq 4347

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

Assertion

Ref	Expression
iotaeq	⊢ (∀x x = y → (℩xφ) = (℩yφ))

Proof of Theorem iotaeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drsb1 2022	. . . . . . 7 ⊢ (∀x x = y → ([z / x]φ ↔ [z / y]φ))
2		df-clab 2340	. . . . . . 7 ⊢ (z ∈ {x ∣ φ} ↔ [z / x]φ)
3		df-clab 2340	. . . . . . 7 ⊢ (z ∈ {y ∣ φ} ↔ [z / y]φ)
4	1, 2, 3	3bitr4g 279	. . . . . 6 ⊢ (∀x x = y → (z ∈ {x ∣ φ} ↔ z ∈ {y ∣ φ}))
5	4	eqrdv 2351	. . . . 5 ⊢ (∀x x = y → {x ∣ φ} = {y ∣ φ})
6	5	eqeq1d 2361	. . . 4 ⊢ (∀x x = y → ({x ∣ φ} = {z} ↔ {y ∣ φ} = {z}))
7	6	abbidv 2467	. . 3 ⊢ (∀x x = y → {z ∣ {x ∣ φ} = {z}} = {z ∣ {y ∣ φ} = {z}})
8	7	unieqd 3902	. 2 ⊢ (∀x x = y → ∪{z ∣ {x ∣ φ} = {z}} = ∪{z ∣ {y ∣ φ} = {z}})
9		df-iota 4339	. 2 ⊢ (℩xφ) = ∪{z ∣ {x ∣ φ} = {z}}
10		df-iota 4339	. 2 ⊢ (℩yφ) = ∪{z ∣ {y ∣ φ} = {z}}
11	8, 9, 10	3eqtr4g 2410	1 ⊢ (∀x x = y → (℩xφ) = (℩yφ))

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  {csn 3737  ∪cuni 3891  ℩cio 4337

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 2478  df-rex 2620  df-uni 3892  df-iota 4339

This theorem is referenced by: (None)