Theorem rspcedeq2vd 3598

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3592 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 3595	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096

This theorem is referenced by:  elpr2elpr  4838  fsetfocdm  8857  symgextfo  19491  smatvscl  22649  eucrctshift  30534  fimgmcyc  43193  ntrclsneine0lem  44681  mogoldbblem  48373  sbgoldbwt  48430  sbgoldbo  48440