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Theorem spsbbi 1722
Description: Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
spsbbi (x(φψ) → ([y / x]φ ↔ [y / x]ψ))

Proof of Theorem spsbbi
StepHypRef Expression
1 spsbim 1721 . . 3 (x(φψ) → ([y / x]φ → [y / x]ψ))
2 spsbim 1721 . . 3 (x(ψφ) → ([y / x]ψ → [y / x]φ))
31, 2anim12i 321 . 2 ((x(φψ) x(ψφ)) → (([y / x]φ → [y / x]ψ) ([y / x]ψ → [y / x]φ)))
4 albiim 1373 . 2 (x(φψ) ↔ (x(φψ) x(ψφ)))
5 dfbi2 368 . 2 (([y / x]φ ↔ [y / x]ψ) ↔ (([y / x]φ → [y / x]ψ) ([y / x]ψ → [y / x]φ)))
63, 4, 53imtr4i 190 1 (x(φψ) → ([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-4 1397  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-sb 1643
This theorem is referenced by:  sbbid  1723  relelfvdm  5148
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