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Theorem spimh 1622
Description: Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1623 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
spimh.1 (ψxψ)
spimh.2 (x = y → (φψ))
Assertion
Ref Expression
spimh (xφψ)

Proof of Theorem spimh
StepHypRef Expression
1 spimh.2 . . . 4 (x = y → (φψ))
2 spimh.1 . . . 4 (ψxψ)
31, 2syl6com 31 . . 3 (φ → (x = yxψ))
43alimi 1341 . 2 (xφx(x = yxψ))
5 ax9o 1585 . 2 (x(x = yxψ) → ψ)
64, 5syl 14 1 (xφψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240
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-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  spim  1623
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