Theorem reueq1 3140

Description: Equality theorem for restricted uniqueness 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 2764	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2764	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	reueq1f 3136	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  ∃!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-cleq 2615  df-clel 2618  df-nfc 2753  df-reu 2919

This theorem is referenced by:  reueqd  3148  lubfval  16978  glbfval  16991  uspgredg2vlem  26115  uspgredg2v  26116  isfrgr  27122  frgr1v  27135  nfrgr2v  27136  frgr3v  27139  1vwmgr  27140  3vfriswmgr  27142  isplig  27328  hdmap14lem4a  37163  hdmap14lem15  37174