Theorem rspcdv2 3573

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 3570	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	1, 4	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053

This theorem is referenced by:  frd  5589  ufdprmidl  33633  cantnf2  43676  imo72b2  44522