Theorem rspcdv2 3630

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 3627	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	1, 4	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068

This theorem is referenced by:  frd  5656  ufdprmidl  33534  cantnf2  43287  imo72b2  44134