Theorem rspcedv 3453

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 238	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
4	1, 3	rspcimedv 3451	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342

This theorem is referenced by:  rspcebdv  3454  rspcedvd  3456  fsuppmapnn0fiubOLD  13005  wrdl1exs1  13604  0csh0  13759  gcdcllem1  15443  nn0gsumfz  18600  pmatcollpw3lem  20810  pmatcollpw3fi1lem2  20814  pm2mpfo  20841  f1otrg  25971  cusgrfilem2  26583  wwlksnredwwlkn  27034  wwlksnextprop  27051  clwwlknun  27282  clwwlknunOLD  27286  cusconngr  27364  xrofsup  29863  esum2d  30485  rexzrexnn0  37888  ov2ssiunov2  38512  lcoel0  42745  lcoss  42753  el0ldep  42783  ldepspr  42790  islindeps2  42800  isldepslvec2  42802