Theorem rspcedeq2vd 3614

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3608 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)

Hypotheses

Ref	Expression
rspcedeqvd.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcedeqvd.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)

Assertion

Ref	Expression
rspcedeq2vd	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

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

Allowed substitution hint:   𝐷(𝑥)

Proof of Theorem rspcedeq2vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspcedeqvd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
3	2	eqcomd 2732	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐶)
4	3	eqeq2d 2737	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐶 = 𝐶))
5		eqidd 2727	. 2 ⊢ (𝜑 → 𝐶 = 𝐶)
6	1, 4, 5	rspcedvd 3608	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065

This theorem is referenced by:  symgextfo  19342  smatvscl  22381  eucrctshift  30005  ntrclsneine0lem  43391  mogoldbblem  46960  sbgoldbwt  47017  sbgoldbo  47027