Theorem reueq1 3415

Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2139, ax-11 2155, and ax-12 2175. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2108. (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 3320	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑))
2		rmoeq1 3414	. . 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 3380	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
5		reu5 3380	. 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 1537  ∃wrex 3068  ∃!wreu 3376  ∃*wrmo 3377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-eu 2567  df-cleq 2727  df-rex 3069  df-rmo 3378  df-reu 3379

This theorem is referenced by:  reueqd  3419  reueqdv  3420  lubfval  18408  glbfval  18421  uspgredg2vlem  29255  uspgredg2v  29256  isfrgr  30289  frgr1v  30300  nfrgr2v  30301  frgr3v  30304  1vwmgr  30305  3vfriswmgr  30307  isplig  30505  hdmap14lem4a  41854  hdmap14lem15  41865