Theorem rspcedv 3571

Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
rspcdv.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcdv.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rspcedv	⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcedv

Step	Hyp	Ref	Expression
1		rspcdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspcdv.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3	2	biimprd 248	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
4	1, 3	rspcimedv 3569	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063

This theorem is referenced by:  rspcebdv  3572  rspcev  3578  rspcedvd  3580  0csh0  14728  gcdcllem1  16438  nn0gsumfz  19925  pmatcollpw3lem  22739  pmatcollpw3fi1lem2  22743  pm2mpfo  22770  f1otrg  28955  cusgrfilem2  29542  wwlksnredwwlkn  29980  wwlksnextprop  29997  clwwlknun  30199  cusconngr  30278  xrofsup  32858  esum2d  34271  rexzrexnn0  43161  onsucelab  43620  ordnexbtwnsuc  43624  ov2ssiunov2  44056  requad2  47983  lcoel0  48788  lcoss  48796  el0ldep  48826  ldepspr  48833  islindeps2  48843  isldepslvec2  48845  affinecomb1  49062  isisod  49386