Theorem iotajust 5179

Description: Soundness justification theorem for df-iota 5180. (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 3605	. . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
2	1	eqeq2d 2189	. . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤}))
3	2	cbvabv 2302	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
4		sneq 3605	. . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧})
5	4	eqeq2d 2189	. . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧}))
6	5	cbvabv 2302	. . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
7	3, 6	eqtri 2198	. 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
8	7	unieqi 3821	1 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}

Syntax hints:   = wceq 1353  {cab 2163  {csn 3594  ∪ cuni 3811

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-sn 3600  df-uni 3812

This theorem is referenced by: (None)