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Theorem sbcom2v2 1859
Description: Lemma for proving sbcom2 1860. It is the same as sbcom2v 1858 but removes the distinct variable constraint on x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbcom2v2 ([w / z][y / x]φ ↔ [y / x][w / z]φ)
Distinct variable groups:   x,w,z   y,z
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem sbcom2v2
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 sbcom2v 1858 . . 3 ([w / z][y / v][v / x]φ ↔ [y / v][w / z][v / x]φ)
2 sbcom2v 1858 . . . 4 ([w / z][v / x]φ ↔ [v / x][w / z]φ)
32sbbii 1645 . . 3 ([y / v][w / z][v / x]φ ↔ [y / v][v / x][w / z]φ)
41, 3bitri 173 . 2 ([w / z][y / v][v / x]φ ↔ [y / v][v / x][w / z]φ)
5 ax-17 1416 . . . 4 (φvφ)
65sbco2v 1818 . . 3 ([y / v][v / x]φ ↔ [y / x]φ)
76sbbii 1645 . 2 ([w / z][y / v][v / x]φ ↔ [w / z][y / x]φ)
8 ax-17 1416 . . 3 ([w / z]φv[w / z]φ)
98sbco2v 1818 . 2 ([y / v][v / x][w / z]φ ↔ [y / x][w / z]φ)
104, 7, 93bitr3i 199 1 ([w / z][y / x]φ ↔ [y / x][w / z]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [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:  sbcom2  1860
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