Theorem rspcedv 3576

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 251	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
4	1, 3	rspcimedv 3574	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092

This theorem is used by:  rspcebdv  3577  rspcev  3583  rspcedvd  3585  0csh0  14854  gcdcllem1  16579  nn0gsumfz  20098  pmatcollpw3lem  22990  pmatcollpw3fi1lem2  22994  pm2mpfo  23021  f1otrg  29275  cusgrfilem2  29864  wwlksnredwwlkn  30311  wwlksnextprop  30328  clwwlknun  30530  cusconngr  30613  xrofsup  33182  esum2d  34547  rexzrexnn0  43589  onsucelab  44048  ordnexbtwnsuc  44052  ov2ssiunov2  44484  requad2  48446  lcoel0  49265  lcoss  49273  el0ldep  49303  ldepspr  49310  islindeps2  49320  isldepslvec2  49322  affinecomb1  49539  isisod  49862