Theorem aiotajust 47821

Description: Soundness justification theorem for df-aiota 47822. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
aiotajust	⊢ ∩ {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 aiotajust

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4599	. . . . 5 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
2	1	eqeq2d 2774	. . . 4 ⊢ (𝑦 = 𝑤 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑤}))
3	2	cbvabv 2833	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}}
4		sneq 4599	. . . . 5 ⊢ (𝑤 = 𝑧 → {𝑤} = {𝑧})
5	4	eqeq2d 2774	. . . 4 ⊢ (𝑤 = 𝑧 → ({𝑥 ∣ 𝜑} = {𝑤} ↔ {𝑥 ∣ 𝜑} = {𝑧}))
6	5	cbvabv 2833	. . 3 ⊢ {𝑤 ∣ {𝑥 ∣ 𝜑} = {𝑤}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
7	3, 6	eqtri 2786	. 2 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}
8	7	inteqi 4916	1 ⊢ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}}

Syntax hints:   = wceq 1570  {cab 2741  {csn 4589  ∩ cint 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-ral 3080  df-rex 3090  df-sn 4590  df-int 4913

This theorem is referenced by: (None)