ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iotajust GIF version

Theorem iotajust 4966
Description: Soundness justification theorem for df-iota 4967. (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 3452 . . . . 5 (𝑦 = 𝑤 → {𝑦} = {𝑤})
21eqeq2d 2099 . . . 4 (𝑦 = 𝑤 → ({𝑥𝜑} = {𝑦} ↔ {𝑥𝜑} = {𝑤}))
32cbvabv 2211 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑤 ∣ {𝑥𝜑} = {𝑤}}
4 sneq 3452 . . . . 5 (𝑤 = 𝑧 → {𝑤} = {𝑧})
54eqeq2d 2099 . . . 4 (𝑤 = 𝑧 → ({𝑥𝜑} = {𝑤} ↔ {𝑥𝜑} = {𝑧}))
65cbvabv 2211 . . 3 {𝑤 ∣ {𝑥𝜑} = {𝑤}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
73, 6eqtri 2108 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
87unieqi 3658 1 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑧 ∣ {𝑥𝜑} = {𝑧}}
Colors of variables: wff set class
Syntax hints:   = wceq 1289  {cab 2074  {csn 3441   cuni 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-sn 3447  df-uni 3649
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator