Theorem rspceeqv 2948

Description: Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)

Hypothesis

Ref	Expression
rspceeqv.1	⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)

Assertion

Ref	Expression
rspceeqv	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶)

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

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem rspceeqv

Step	Hyp	Ref	Expression
1		rspceeqv.1	. . 3 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷)
2	1	eqeq2d 2250	. 2 ⊢ (𝑥 = 𝐴 → (𝐸 = 𝐶 ↔ 𝐸 = 𝐷))
3	2	rspcev 2929	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∃wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823

This theorem is used by:  elixpsn  7021  ixpsnf1o  7022  elfir  7312  0ct  7453  ctmlemr  7454  ctssdclemn0  7456  fodju0  7493  ccats1pfxeqrex  11509  mertenslemi1  12327  mertenslem2  12328  nninfctlemfo  12842  pcprmpw  13142  1arithlem4  13174  ctiunctlemfo  13388  elrestr  13660  lss1d  14829  lspsn  14862  znf1o  15095  restopnb  15407  mopnex  15731  metrest  15732  mpodvdsmulf1o  16284  lgsquadlem1  16401  2sqlem2  16439  mul2sq  16440  2sqlem3  16441  2sqlem9  16448  2sqlem10  16449  nnnninfex  17271