Theorem rspcimdv 3562

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
rspcimdv	⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcimdv

Step	Hyp	Ref	Expression
1		df-ral 3048	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
2		rspcimdv.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
4	3	eleq1d 2816	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
5	4	biimprd 248	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵))
6		rspcimdv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
7	5, 6	imim12d 81	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒)))
8	2, 7	spcimdv 3543	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒)))
9	2, 8	mpid 44	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → 𝜒))
10	1, 9	biimtrid 242	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048

This theorem is referenced by:  rspcimedv  3563  rspcdv  3564  wrd2ind  14630  mreexd  17548  mreexexlemd  17550  catcocl  17591  catass  17592  moni  17643  subccocl  17752  funcco  17778  fullfo  17821  fthf1  17826  nati  17865  acsfiindd  18459  chpscmat  22757  mpomulcn  24785  friendshipgt3  30378  lmxrge0  33965  funressnfv  47142  cfsetsnfsetf1  47158