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Theorem r19.23t 2728
 Description: Closed theorem form of r19.23 2729. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
r19.23t (Ⅎxψ → (x A (φψ) ↔ (x A φψ)))

Proof of Theorem r19.23t
StepHypRef Expression
1 19.23t 1800 . 2 (Ⅎxψ → (x((x A φ) → ψ) ↔ (x(x A φ) → ψ)))
2 df-ral 2619 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
3 impexp 433 . . . 4 (((x A φ) → ψ) ↔ (x A → (φψ)))
43albii 1566 . . 3 (x((x A φ) → ψ) ↔ x(x A → (φψ)))
52, 4bitr4i 243 . 2 (x A (φψ) ↔ x((x A φ) → ψ))
6 df-rex 2620 . . 3 (x A φx(x A φ))
76imbi1i 315 . 2 ((x A φψ) ↔ (x(x A φ) → ψ))
81, 5, 73bitr4g 279 1 (Ⅎxψ → (x A (φψ) ↔ (x A φψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620 This theorem is referenced by:  r19.23  2729  rexlimd2  2736
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