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Theorem sbco2vlem 1817
Description: This is a version of sbco2 1836 where z is distinct from x and from y. It is a lemma on the way to proving sbco2v 1818 which only requires that z and x be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) (One distinct variable constraint removed by Jim Kingdon, 3-Feb-2018.)
Hypothesis
Ref Expression
sbco2vlem.1 (φzφ)
Assertion
Ref Expression
sbco2vlem ([y / z][z / x]φ ↔ [y / x]φ)
Distinct variable groups:   x,z   y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbco2vlem
StepHypRef Expression
1 sbco2vlem.1 . . 3 (φzφ)
21hbsbv 1814 . 2 ([y / x]φz[y / x]φ)
3 sbequ 1718 . 2 (z = y → ([z / x]φ ↔ [y / x]φ))
42, 3sbieh 1670 1 ([y / z][z / x]φ ↔ [y / x]φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240  [wsb 1642
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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sbco2v  1818
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