Theorem reueq1 3391

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2142, ax-11 2158, and ax-12 2178. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2111. (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 3297	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
2		rmoeq1 3390	. . 3 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
3	1, 2	anbi12d 632	. 2 ⊢ (𝐴 = 𝐵 → ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑)))
4		reu5 3358	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
5		reu5 3358	. 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 206   ∧ wa 395   = wceq 1540  ∃wrex 3054  ∃!wreu 3354  ∃*wrmo 3355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2534  df-eu 2563  df-cleq 2722  df-rex 3055  df-rmo 3356  df-reu 3357

This theorem is referenced by:  reueqd  3395  reueqdv  3396  lubfval  18316  glbfval  18329  uspgredg2vlem  29157  uspgredg2v  29158  isfrgr  30196  frgr1v  30207  nfrgr2v  30208  frgr3v  30211  1vwmgr  30212  3vfriswmgr  30214  isplig  30412  hdmap14lem4a  41872  hdmap14lem15  41883