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Theorem rexlimd2 3256
Description: Version of rexlimd 3257 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 3248 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
4 rexlimd2.2 . . 3 (𝜑 → Ⅎ𝑥𝜒)
5 r19.23t 3246 . . 3 (Ⅎ𝑥𝜒 → (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒)))
64, 5syl 17 . 2 (𝜑 → (∀𝑥𝐴 (𝜓𝜒) ↔ (∃𝑥𝐴 𝜓𝜒)))
73, 6mpbid 231 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wnf 1777  wcel 2098  wral 3055  wrex 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-ral 3056  df-rex 3065
This theorem is referenced by:  rexlimd  3257  sbcrext  3862
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