Theorem reueq1 2731

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
reueq1	⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reueq1

Step	Hyp	Ref	Expression
1		nfcv 2373	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2373	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	reueq1f 2727	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  ∃!wreu 2511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-cleq 2223  df-clel 2226  df-nfc 2362  df-reu 2516

This theorem is referenced by:  reueqd  2743  ringideu  14054  uspgredg2vlem  16100  uspgredg2v  16101