Theorem rspcedeq2vd 3458

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3456 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 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐶)
4	3	eqeq2d 2770	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐶 = 𝐶))
5		eqidd 2761	. 2 ⊢ (𝜑 → 𝐶 = 𝐶)
6	1, 4, 5	rspcedvd 3456	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342

This theorem is referenced by:  symgextfo  18062  smatvscl  20552  eucrctshift  27416  ntrclsneine0lem  38882  mogoldbblem  42157  sbgoldbwt  42193  sbgoldbo  42203