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Theorem aiotajust 44248
Description: Soundness justification theorem for df-aiota 44249. (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
aiotajust {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {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.
StepHypRef Expression
1 sneq 4551 . . . . 5 (𝑦 = 𝑤 → {𝑦} = {𝑤})
21eqeq2d 2748 . . . 4 (𝑦 = 𝑤 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑤}))
32cbvabv 2811 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
4 sneq 4551 . . . . 5 (𝑤 = 𝑧 → {𝑤} = {𝑧})
54eqeq2d 2748 . . . 4 (𝑤 = 𝑧 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑧}))
65cbvabv 2811 . . 3 {𝑤 ∣ {𝑥𝜑} = {𝑤}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
73, 6eqtri 2765 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
87inteqi 4863 1 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  {cab 2714  {csn 4541   cint 4859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-ral 3066  df-sn 4542  df-int 4860
This theorem is referenced by: (None)
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