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Theorem r19.37 2760
 Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
r19.37.1 xφ
Assertion
Ref Expression
r19.37 (x A (φψ) → (φx A ψ))

Proof of Theorem r19.37
StepHypRef Expression
1 r19.35 2758 . 2 (x A (φψ) ↔ (x A φx A ψ))
2 r19.37.1 . . . 4 xφ
3 ax-1 6 . . . 4 (φ → (x Aφ))
42, 3ralrimi 2695 . . 3 (φx A φ)
54imim1i 54 . 2 ((x A φx A ψ) → (φx A ψ))
61, 5sylbi 187 1 (x A (φψ) → (φx A ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620 This theorem is referenced by:  r19.37av  2761
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