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Theorem sbi2v 1769
 Description: Reverse direction of sbimv 1770. (Contributed by Jim Kingdon, 18-Jan-2018.)
Assertion
Ref Expression
sbi2v (([y / x]φ → [y / x]ψ) → [y / x](φψ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem sbi2v
StepHypRef Expression
1 19.38 1563 . . 3 ((x(x = y φ) → x(x = yψ)) → x((x = y φ) → (x = yψ)))
2 pm3.3 248 . . . . 5 (((x = y φ) → (x = yψ)) → (x = y → (φ → (x = yψ))))
3 pm2.04 76 . . . . 5 ((φ → (x = yψ)) → (x = y → (φψ)))
42, 3syli 33 . . . 4 (((x = y φ) → (x = yψ)) → (x = y → (φψ)))
54alimi 1341 . . 3 (x((x = y φ) → (x = yψ)) → x(x = y → (φψ)))
61, 5syl 14 . 2 ((x(x = y φ) → x(x = yψ)) → x(x = y → (φψ)))
7 sb5 1764 . . 3 ([y / x]φx(x = y φ))
8 sb6 1763 . . 3 ([y / x]ψx(x = yψ))
97, 8imbi12i 228 . 2 (([y / x]φ → [y / x]ψ) ↔ (x(x = y φ) → x(x = yψ)))
10 sb6 1763 . 2 ([y / x](φψ) ↔ x(x = y → (φψ)))
116, 9, 103imtr4i 190 1 (([y / x]φ → [y / x]ψ) → [y / x](φψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃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-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  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-sb 1643 This theorem is referenced by:  sbimv  1770
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