Theorem reueq1 3411

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2130, ax-11 2147, and ax-12 2167. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2101. (Revised by Wolf Lammen, 12-Mar-2025.)

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		rexeq 3317	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
2		rmoeq1 3410	. . 3 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
3	1, 2	anbi12d 631	. 2 ⊢ (𝐴 = 𝐵 → ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑)))
4		reu5 3374	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
5		reu5 3374	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  ∃wrex 3066  ∃!wreu 3370  ∃*wrmo 3371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-mo 2530  df-eu 2559  df-cleq 2720  df-rex 3067  df-rmo 3372  df-reu 3373

This theorem is referenced by:  reueqd  3415  lubfval  18335  glbfval  18348  uspgredg2vlem  29029  uspgredg2v  29030  isfrgr  30063  frgr1v  30074  nfrgr2v  30075  frgr3v  30078  1vwmgr  30079  3vfriswmgr  30081  isplig  30279  hdmap14lem4a  41338  hdmap14lem15  41349