Theorem rspcdv2 3579

Description: Restricted specialization, using implicit substitution. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
rspcdv2.1	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
rspcdv2.2	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcdv2.3	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)

Assertion

Ref	Expression
rspcdv2	⊢ (𝜑 → 𝜒)

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdv2

Step	Hyp	Ref	Expression
1		rspcdv2.3	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
2		rspcdv2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		rspcdv2.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	2, 3	rspcdv 3576	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	1, 4	mpd 16	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080

This theorem is referenced by:  frd  5608  ufdprmidl  33743  cantnf2  43909  imo72b2  44755