Theorem rspcedeqvd 3587

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3582 for equations. (Contributed by AV, 24-Dec-2019.)

Hypotheses

Ref	Expression
rspcedeqvd.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcedeqvd.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)

Assertion

Ref	Expression
rspcedeqvd	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

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

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

Proof of Theorem rspcedeqvd

Step	Hyp	Ref	Expression
1		rspcedeqvd.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
2		rspcedeqvd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3	1, 2	rspcime 3585	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091

This theorem is used by:  elpr2elpr  4827  fsetfocdm  8908  mod2eq1n2dvds  16543  symgextfo  19663  fincygsubgodexd  20356  smatvscl  22866  eucrctshift  30871  fsuppcurry1  33383  fsuppcurry2  33384  nnn1suc  43522  fimgmcyc  43789  ntrclsneine0lem  45254  mogoldbblem  49021  sbgoldbwt  49078  sbgoldbo  49088