Theorem reueq1 3366

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2111, ax-11 2125, and ax-12 2140. (Revised by Steven Nguyen, 30-Apr-2023.)

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		eleq2 2870	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	anbi1d 629	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	eubidv 2631	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
4		df-reu 3111	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5		df-reu 3111	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
6	3, 4, 5	3bitr4g 315	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2080  ∃!weu 2610  ∃!wreu 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-ext 2768

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1763  df-mo 2575  df-eu 2611  df-cleq 2787  df-clel 2862  df-reu 3111

This theorem is referenced by:  reueqd  3378  lubfval  17417  glbfval  17430  uspgredg2vlem  26688  uspgredg2v  26689  isfrgr  27721  frgr1v  27734  nfrgr2v  27735  frgr3v  27738  1vwmgr  27739  3vfriswmgr  27741  isplig  27936  hdmap14lem4a  38551  hdmap14lem15  38562