Theorem rspcedv 2788

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 157	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
4	1, 3	rspcimedv 2786	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∃wrex 2415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683

This theorem is referenced by:  rspcedvd  2790  rexxfrd  4379  enomnilem  7003  ltexnqq  7209  halfnqq  7211  ltbtwnnqq  7216  genpml  7318  genpmu  7319  genprndl  7322  genprndu  7323  axarch  7692  apreap  8342