Theorem rspcedeq1vd 3614

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3611 for equations, in which the left 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
rspcedeq1vd	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

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

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem rspcedeq1vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspcedeqvd.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
3	2	eqeq1d 2733	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐷 = 𝐷))
4		eqidd 2732	. 2 ⊢ (𝜑 → 𝐷 = 𝐷)
5	1, 3, 4	rspcedvd 3611	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070

This theorem is referenced by:  mod2eq1n2dvds  16272  fincygsubgodexd  19942  fsuppcurry1  31821  fsuppcurry2  31822