Theorem rspcdvinvd 38474

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 3312	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))
5	1, 4	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202

This theorem is referenced by:  imo72b2  38475