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Theorem spsbim 1721
Description: Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
spsbim (x(φψ) → ([y / x]φ → [y / x]ψ))

Proof of Theorem spsbim
StepHypRef Expression
1 imim2 49 . . . 4 ((φψ) → ((x = yφ) → (x = yψ)))
21sps 1427 . . 3 (x(φψ) → ((x = yφ) → (x = yψ)))
3 id 19 . . . . . 6 ((φψ) → (φψ))
43anim2d 320 . . . . 5 ((φψ) → ((x = y φ) → (x = y ψ)))
54alimi 1341 . . . 4 (x(φψ) → x((x = y φ) → (x = y ψ)))
6 exim 1487 . . . 4 (x((x = y φ) → (x = y ψ)) → (x(x = y φ) → x(x = y ψ)))
75, 6syl 14 . . 3 (x(φψ) → (x(x = y φ) → x(x = y ψ)))
82, 7anim12d 318 . 2 (x(φψ) → (((x = yφ) x(x = y φ)) → ((x = yψ) x(x = y ψ))))
9 df-sb 1643 . 2 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
10 df-sb 1643 . 2 ([y / x]ψ ↔ ((x = yψ) x(x = y ψ)))
118, 9, 103imtr4g 194 1 (x(φψ) → ([y / x]φ → [y / x]ψ))
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-4 1397  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  spsbbi  1722  hbsb4t  1886  moim  1961
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