Theorem iotajust 4338

Description: Soundness justification theorem for df-iota 4339. (Contributed by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
iotajust	⊢ ∪{y ∣ {x ∣ φ} = {y}} = ∪{z ∣ {x ∣ φ} = {z}}

Distinct variable groups:   x,z   φ,z   φ,y   x,y

Allowed substitution hint:   φ(x)

Proof of Theorem iotajust

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3744	. . . . 5 ⊢ (y = w → {y} = {w})
2	1	eqeq2d 2364	. . . 4 ⊢ (y = w → ({x ∣ φ} = {y} ↔ {x ∣ φ} = {w}))
3	2	cbvabv 2472	. . 3 ⊢ {y ∣ {x ∣ φ} = {y}} = {w ∣ {x ∣ φ} = {w}}
4		sneq 3744	. . . . 5 ⊢ (w = z → {w} = {z})
5	4	eqeq2d 2364	. . . 4 ⊢ (w = z → ({x ∣ φ} = {w} ↔ {x ∣ φ} = {z}))
6	5	cbvabv 2472	. . 3 ⊢ {w ∣ {x ∣ φ} = {w}} = {z ∣ {x ∣ φ} = {z}}
7	3, 6	eqtri 2373	. 2 ⊢ {y ∣ {x ∣ φ} = {y}} = {z ∣ {x ∣ φ} = {z}}
8	7	unieqi 3901	1 ⊢ ∪{y ∣ {x ∣ φ} = {y}} = ∪{z ∣ {x ∣ φ} = {z}}

Syntax hints:   = wceq 1642  {cab 2339  {csn 3737  ∪cuni 3891

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-sn 3741  df-uni 3892

This theorem is referenced by: (None)