Theorem rspcda 3315

Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 29-Jun-2020.)

Hypotheses

Ref	Expression
rspcdva.1	⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒))
rspcdva.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
rspcdva.3	⊢ (𝜑 → 𝐶 ∈ 𝐴)
rspcda.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
rspcda	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rspcda

Step	Hyp	Ref	Expression
1		rspcdva.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
2		rspcdva.2	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
3		rspcdva.1	. . 3 ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒))
4	3	rspcv 3305	. 2 ⊢ (𝐶 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜓 → 𝜒))
5	1, 2, 4	sylc 65	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  Ⅎwnf 1708   ∈ 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: (None)