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Theorem r19.45av 2464
 Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.45av (x A (φ ψ) → (φ x A ψ))
Distinct variable group:   φ,x
Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem r19.45av
StepHypRef Expression
1 r19.43 2462 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
2 idd 21 . . . 4 (x A → (φφ))
32rexlimiv 2421 . . 3 (x A φφ)
43orim1i 676 . 2 ((x A φ x A ψ) → (φ x A ψ))
51, 4sylbi 114 1 (x A (φ ψ) → (φ x A ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 628   ∈ 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-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305  df-rex 2306 This theorem is referenced by: (None)
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