Theorem rspcedeq2vd 3559

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3555 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 2744	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐶)
4	3	eqeq2d 2749	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐶 = 𝐶))
5		eqidd 2739	. 2 ⊢ (𝜑 → 𝐶 = 𝐶)
6	1, 4, 5	rspcedvd 3555	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069

This theorem is referenced by:  symgextfo  18945  smatvscl  21581  eucrctshift  28508  ntrclsneine0lem  41563  mogoldbblem  45060  sbgoldbwt  45117  sbgoldbo  45127