Theorem reueq1 3117

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 2751	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2751	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	reueq1f 3113	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃!wreu 2898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-cleq 2603  df-clel 2606  df-nfc 2740  df-reu 2903

This theorem is referenced by:  reueqd  3125  lubfval  16801  glbfval  16814  isfrgra  26517  frgra3v  26529  1vwmgra  26530  3vfriswmgra  26532  isplig  26720  hdmap14lem4a  36181  hdmap14lem15  36192  uspgredg2vlem  40450  uspgredg2v  40451  frgr1v  41441  nfrgr2v  41442  frgr3v  41445  1vwmgr  41446  3vfriswmgr  41448