Theorem iotajust 5285

Description: Soundness justification theorem for df-iota 5286. (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 3680	. . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
2	1	eqeq2d 2243	. . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤}))
3	2	cbvabv 2356	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
4		sneq 3680	. . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧})
5	4	eqeq2d 2243	. . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧}))
6	5	cbvabv 2356	. . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
7	3, 6	eqtri 2252	. 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
8	7	unieqi 3903	1 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}

Syntax hints:   = wceq 1397  {cab 2217  {csn 3669  ∪ cuni 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-sn 3675  df-uni 3894

This theorem is referenced by: (None)