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Theorem r19.23t 2440
Description: Closed theorem form of r19.23 2441. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Assertion
Ref Expression
r19.23t (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Proof of Theorem r19.23t
StepHypRef Expression
1 19.23t 1583 . 2 (Ⅎ𝑥𝜓 → (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓)))
2 df-ral 2328 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
3 impexp 254 . . . 4 (((𝑥𝐴𝜑) → 𝜓) ↔ (𝑥𝐴 → (𝜑𝜓)))
43albii 1375 . . 3 (∀𝑥((𝑥𝐴𝜑) → 𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
52, 4bitr4i 180 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → 𝜓))
6 df-rex 2329 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
76imbi1i 231 . 2 ((∃𝑥𝐴 𝜑𝜓) ↔ (∃𝑥(𝑥𝐴𝜑) → 𝜓))
81, 5, 73bitr4g 216 1 (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wnf 1365  wex 1397  wcel 1409  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-ral 2328  df-rex 2329
This theorem is referenced by:  r19.23  2441  rexlimd2  2448
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