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Theorem 19.21h 1422
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." New proofs should use 19.21 1448 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.21h.1 (φxφ)
Assertion
Ref Expression
19.21h (x(φψ) ↔ (φxψ))

Proof of Theorem 19.21h
StepHypRef Expression
1 19.21h.1 . . 3 (φxφ)
2 alim 1319 . . 3 (x(φψ) → (xφxψ))
31, 2syl5 28 . 2 (x(φψ) → (φxψ))
4 hba1 1406 . . . 4 (xψxxψ)
51, 4hbim 1410 . . 3 ((φxψ) → x(φxψ))
6 ax-4 1373 . . . 4 (xψψ)
76imim2i 12 . . 3 ((φxψ) → (φψ))
85, 7alrimih 1331 . 2 ((φxψ) → x(φψ))
93, 8impbii 117 1 (x(φψ) ↔ (φxψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 100  ax-ia3 101  ax-5 1309  ax-gen 1311  ax-4 1373  ax-ial 1400  ax-i5r 1401
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  hbim1  1435  nf3  1532  19.21v  1726
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