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Theorem sborv 1767
Description: Version of sbor 1825 where x and y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
Assertion
Ref Expression
sborv ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem sborv
StepHypRef Expression
1 sb5 1764 . . 3 ([y / x](φ ψ) ↔ x(x = y (φ ψ)))
2 andi 730 . . . 4 ((x = y (φ ψ)) ↔ ((x = y φ) (x = y ψ)))
32exbii 1493 . . 3 (x(x = y (φ ψ)) ↔ x((x = y φ) (x = y ψ)))
4 19.43 1516 . . 3 (x((x = y φ) (x = y ψ)) ↔ (x(x = y φ) x(x = y ψ)))
51, 3, 43bitri 195 . 2 ([y / x](φ ψ) ↔ (x(x = y φ) x(x = y ψ)))
6 sb5 1764 . . 3 ([y / x]φx(x = y φ))
7 sb5 1764 . . 3 ([y / x]ψx(x = y ψ))
86, 7orbi12i 680 . 2 (([y / x]φ [y / x]ψ) ↔ (x(x = y φ) x(x = y ψ)))
95, 8bitr4i 176 1 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wo 628  wex 1378  [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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  sbor  1825
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