Theorem rexlimi 3243

Description: Restricted quantifier version of exlimi 2213. (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
rexlimi.1	⊢ Ⅎ𝑥𝜓
rexlimi.2	⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
rexlimi	⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Proof of Theorem rexlimi

Step	Hyp	Ref	Expression
1		rexlimi.2	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
2	1	rgen 3073	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)
3		rexlimi.1	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	r19.23 3242	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
5	2, 4	mpbi 229	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1787   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-ral 3068  df-rex 3069

This theorem is referenced by:  reuan  3825  triun  5200  reusv1  5315  reusv3  5323  iunopeqop  5429  tfinds  7681  fiun  7759  f1iun  7760  iunfo  10226  iundom2g  10227  fsumcom2  15414  fprodcom2  15622  dfon2lem7  33671  frpoins3xpg  33714  frpoins3xp3g  33715  nosupbnd1  33844  nosupbnd2  33846  noinfbnd1  33859  noinfbnd2  33861  finminlem  34434  r19.36vf  42574  allbutfiinf  42850  infxrunb3rnmpt  42858  hoidmvlelem1  44023  2zrngmmgm  45392