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Theorem iotajust 4338
 Description: Soundness justification theorem for df-iota 4339. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
iotajust {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 iotajust
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 sneq 3744 . . . . 5 (y = w → {y} = {w})
21eqeq2d 2364 . . . 4 (y = w → ({x φ} = {y} ↔ {x φ} = {w}))
32cbvabv 2472 . . 3 {y {x φ} = {y}} = {w {x φ} = {w}}
4 sneq 3744 . . . . 5 (w = z → {w} = {z})
54eqeq2d 2364 . . . 4 (w = z → ({x φ} = {w} ↔ {x φ} = {z}))
65cbvabv 2472 . . 3 {w {x φ} = {w}} = {z {x φ} = {z}}
73, 6eqtri 2373 . 2 {y {x φ} = {y}} = {z {x φ} = {z}}
87unieqi 3901 1 {y {x φ} = {y}} = {z {x φ} = {z}}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  {cab 2339  {csn 3737  ∪cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-sn 3741  df-uni 3892 This theorem is referenced by: (None)
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