Theorem rspcdvinvd 40598

Description: If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

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

Assertion

Ref	Expression
rspcdvinvd	⊢ (𝜑 → 𝜒)

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdvinvd

Step	Hyp	Ref	Expression
1		rspcdvinvd.3	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
2		rspcdvinvd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		rspcdvinvd.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
4	2, 3	rspcdv 3612	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	1, 4	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-cleq 2813  df-clel 2892  df-ral 3142

This theorem is referenced by:  imo72b2  40599