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Theorem sbanv 1766
 Description: Version of sban 1826 where x and y are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
Assertion
Ref Expression
sbanv ([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 sbanv
StepHypRef Expression
1 sb6 1763 . 2 ([y / x](φ ψ) ↔ x(x = y → (φ ψ)))
2 sb6 1763 . . . 4 ([y / x]φx(x = yφ))
3 sb6 1763 . . . 4 ([y / x]ψx(x = yψ))
42, 3anbi12i 433 . . 3 (([y / x]φ [y / x]ψ) ↔ (x(x = yφ) x(x = yψ)))
5 19.26 1367 . . 3 (x((x = yφ) (x = yψ)) ↔ (x(x = yφ) x(x = yψ)))
6 pm4.76 536 . . . 4 (((x = yφ) (x = yψ)) ↔ (x = y → (φ ψ)))
76albii 1356 . . 3 (x((x = yφ) (x = yψ)) ↔ x(x = y → (φ ψ)))
84, 5, 73bitr2i 197 . 2 (([y / x]φ [y / x]ψ) ↔ x(x = y → (φ ψ)))
91, 8bitr4i 176 1 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ 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-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:  sban  1826
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