Theorem reubidva 3404

Description: Formula-building rule for restricted existential uniqueness quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023.)

Hypothesis

Ref	Expression
rmobidva.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
reubidva	⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reubidva

Step	Hyp	Ref	Expression
1		rmobidva.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
2	1	pm5.32da 578	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
3	2	eubidv 2589	. 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
4		df-reu 3389	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		df-reu 3389	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∃!weu 2571  ∃!wreu 3386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543  df-eu 2572  df-reu 3389

This theorem is referenced by:  reubidv  3406  reuxfrd  3770  reuxfr1d  3772  fdmeu  6978  exfo  7139  f1ofveu  7442  zmax  13010  zbtwnre  13011  rebtwnz  13012  icoshftf1o  13534  divalgb  16452  1arith2  16975  ply1divalg2  26198  addsq2reu  27502  addsqn2reu  27503  addsqrexnreu  27504  2sqreultlem  27509  2sqreunnltlem  27512  frgr2wwlkeu  30359  numclwwlk2lem1  30408  numclwlk2lem2f1o  30411  pjhtheu2  31448  reuxfrdf  32519  xrsclat  32994  xrmulc1cn  33876  ply1divalg3  35610  poimirlem25  37605  hdmap14lem14  41838  cantnf2  43287  prproropreud  47383  quad1  47494  requad1  47496  requad2  47497  isuspgrim0lem  47755  isuspgrim0  47756  isuspgrimlem  47758  itscnhlinecirc02p  48519