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Theorem sbhypf 2580
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
sbhypf.1 xψ
sbhypf.2 (x = A → (φψ))
Assertion
Ref Expression
sbhypf (y = A → ([y / x]φψ))
Distinct variable groups:   x,A   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)   A(y)

Proof of Theorem sbhypf
StepHypRef Expression
1 vex 2538 . . 3 y V
2 eqeq1 2028 . . 3 (x = y → (x = Ay = A))
31, 2ceqsexv 2570 . 2 (x(x = y x = A) ↔ y = A)
4 nfs1v 1797 . . . 4 x[y / x]φ
5 sbhypf.1 . . . 4 xψ
64, 5nfbi 1463 . . 3 x([y / x]φψ)
7 sbequ12 1636 . . . . 5 (x = y → (φ ↔ [y / x]φ))
87bicomd 129 . . . 4 (x = y → ([y / x]φφ))
9 sbhypf.2 . . . 4 (x = A → (φψ))
108, 9sylan9bb 438 . . 3 ((x = y x = A) → ([y / x]φψ))
116, 10exlimi 1467 . 2 (x(x = y x = A) → ([y / x]φψ))
123, 11sylbir 125 1 (y = A → ([y / x]φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wnf 1329  wex 1362  [wsb 1627
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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537
This theorem is referenced by:  mob2  2698  tfisi  4237  ralxpf  4409  rexxpf  4410
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