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Theorem r19.23t 2417
 Description: Closed theorem form of r19.23 2418. (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 1564 . 2 (Ⅎxψ → (x((x A φ) → ψ) ↔ (x(x A φ) → ψ)))
2 df-ral 2305 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
3 impexp 250 . . . 4 (((x A φ) → ψ) ↔ (x A → (φψ)))
43albii 1356 . . 3 (x((x A φ) → ψ) ↔ x(x A → (φψ)))
52, 4bitr4i 176 . 2 (x A (φψ) ↔ x((x A φ) → ψ))
6 df-rex 2306 . . 3 (x A φx(x A φ))
76imbi1i 227 . 2 ((x A φψ) ↔ (x(x A φ) → ψ))
81, 5, 73bitr4g 212 1 (Ⅎxψ → (x A (φψ) ↔ (x A φψ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  Ⅎwnf 1346  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  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:  r19.23  2418  rexlimd2  2425
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