Theorem iotajust 6493

Description: Soundness justification theorem for df-iota 6494. (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 4590	. . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
2	1	eqeq2d 2771	. . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤}))
3	2	cbvabv 2830	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
4		sneq 4590	. . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧})
5	4	eqeq2d 2771	. . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧}))
6	5	cbvabv 2830	. . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
7	3, 6	eqtri 2783	. 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
8	7	unieqi 4878	1 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}

This proof depends on syntax axioms:   = wceq 1569  {cab 2738  {csn 4584  ∪ cuni 4866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-sn 4585  df-uni 4867

This theorem is used by: (None)