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Theorem iotajust 4809
Description: Soundness justification theorem for df-iota 4810. (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 3378 . . . . 5 (y = w → {y} = {w})
21eqeq2d 2048 . . . 4 (y = w → ({xφ} = {y} ↔ {xφ} = {w}))
32cbvabv 2158 . . 3 {y ∣ {xφ} = {y}} = {w ∣ {xφ} = {w}}
4 sneq 3378 . . . . 5 (w = z → {w} = {z})
54eqeq2d 2048 . . . 4 (w = z → ({xφ} = {w} ↔ {xφ} = {z}))
65cbvabv 2158 . . 3 {w ∣ {xφ} = {w}} = {z ∣ {xφ} = {z}}
73, 6eqtri 2057 . 2 {y ∣ {xφ} = {y}} = {z ∣ {xφ} = {z}}
87unieqi 3581 1 {y ∣ {xφ} = {y}} = {z ∣ {xφ} = {z}}
Colors of variables: wff set class
Syntax hints:   = wceq 1242  {cab 2023  {csn 3367   cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-sn 3373  df-uni 3572
This theorem is referenced by: (None)
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