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Theorem r19.45mv 3309
 Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.45mv (x x A → (x A (φ ψ) ↔ (φ x A ψ)))
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem r19.45mv
StepHypRef Expression
1 r19.9rmv 3307 . . 3 (x x A → (φx A φ))
21orbi1d 704 . 2 (x x A → ((φ x A ψ) ↔ (x A φ x A ψ)))
3 r19.43 2462 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
42, 3syl6rbbr 188 1 (x x A → (x A (φ ψ) ↔ (φ x A ψ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 628  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301 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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033  df-rex 2306 This theorem is referenced by:  ltexprlemloc  6571
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