Theorem rspcedv 3605

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 247	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
4	1, 3	rspcimedv 3603	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071

This theorem is referenced by:  rspcebdv  3606  rspcev  3612  rspcedvd  3614  0csh0  14747  gcdcllem1  16444  nn0gsumfz  19893  pmatcollpw3lem  22505  pmatcollpw3fi1lem2  22509  pm2mpfo  22536  f1otrg  28377  cusgrfilem2  28968  wwlksnredwwlkn  29404  wwlksnextprop  29421  clwwlknun  29620  cusconngr  29699  xrofsup  32235  esum2d  33377  rexzrexnn0  41844  onsucelab  42315  ordnexbtwnsuc  42319  ov2ssiunov2  42753  requad2  46590  lcoel0  47197  lcoss  47205  el0ldep  47235  ldepspr  47242  islindeps2  47252  isldepslvec2  47254  affinecomb1  47476