Theorem rspcedeq2vd 3584

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3578 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

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

Proof of Theorem rspcedeq2vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
2		rspcedeqvd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3	1, 2	rspcime 3581	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086

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 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087

This theorem is used by:  elpr2elpr  4829  fsetfocdm  8863  symgextfo  19550  smatvscl  22747  eucrctshift  30724  fimgmcyc  43417  ntrclsneine0lem  44905  mogoldbblem  48637  sbgoldbwt  48694  sbgoldbo  48704