Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexlimd2 GIF version

Theorem rexlimd2 2448
 Description: Version of rexlimd 2447 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
rexlimd2.1 𝑥𝜑
rexlimd2.2 (𝜑 → Ⅎ𝑥𝜒)
rexlimd2.3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
Assertion
Ref Expression
rexlimd2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Proof of Theorem rexlimd2
StepHypRef Expression
1 rexlimd2.1 . . 3 𝑥𝜑
2 rexlimd2.3 . . 3 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
31, 2ralrimi 2407 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexlimd2.2 . . 3 (𝜑 → Ⅎ𝑥𝜒)
5 r19.23t 2440 . . 3 (Ⅎ𝑥𝜒 → (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒)))
64, 5syl 14 . 2 (𝜑 → (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒)))
73, 6mpbid 139 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  Ⅎwnf 1365   ∈ 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:  sbcrext  2863
 Copyright terms: Public domain W3C validator