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Theorem iotajust 4894
 Description: Soundness justification theorem for df-iota 4895. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
iotajust {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 iotajust
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sneq 3414 . . . . 5 (𝑦 = 𝑤 → {𝑦} = {𝑤})
21eqeq2d 2067 . . . 4 (𝑦 = 𝑤 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑤}))
32cbvabv 2177 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
4 sneq 3414 . . . . 5 (𝑤 = 𝑧 → {𝑤} = {𝑧})
54eqeq2d 2067 . . . 4 (𝑤 = 𝑧 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑧}))
65cbvabv 2177 . . 3 {𝑤 ∣ {𝑥𝜑} = {𝑤}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
73, 6eqtri 2076 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
87unieqi 3618 1 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
 Colors of variables: wff set class Syntax hints:   = wceq 1259  {cab 2042  {csn 3403  ∪ cuni 3608 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-sn 3409  df-uni 3609 This theorem is referenced by: (None)
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