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Theorem iotajust 5812
 Description: Soundness justification theorem for df-iota 5813. (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 4163 . . . . 5 (𝑦 = 𝑤 → {𝑦} = {𝑤})
21eqeq2d 2636 . . . 4 (𝑦 = 𝑤 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑤}))
32cbvabv 2750 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
4 sneq 4163 . . . . 5 (𝑤 = 𝑧 → {𝑤} = {𝑧})
54eqeq2d 2636 . . . 4 (𝑤 = 𝑧 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑧}))
65cbvabv 2750 . . 3 {𝑤 ∣ {𝑥𝜑} = {𝑤}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
73, 6eqtri 2648 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
87unieqi 4416 1 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480  {cab 2612  {csn 4153  ∪ cuni 4407 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rex 2918  df-sn 4154  df-uni 4408 This theorem is referenced by: (None)
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