Theorem rspcedeq1vd 3590

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3585 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

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem rspcedeq1vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
2		rspcedeqvd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3	1, 2	rspcime 3588	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092

This theorem is used by:  mod2eq1n2dvds  16422  fincygsubgodexd  20208  fsuppcurry1  33098  fsuppcurry2  33099  nnn1suc  43066