Theorem reubii 2718

Description: Formula-building rule for restricted existential quantifier (inference form). (Contributed by NM, 22-Oct-1999.)

Hypothesis

Ref	Expression
reubii.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
reubii	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓)

Proof of Theorem reubii

Step	Hyp	Ref	Expression
1		reubii.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	a1i 9	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	reubiia 2717	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜓)

Syntax hints:   ↔ wb 105   ∈ wcel 2200  ∃!wreu 2510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-eu 2080  df-reu 2515

This theorem is referenced by:  caucvgsrlemcl  7972  axcaucvglemcl  8078  axcaucvglemval  8080  uspgredgiedg  15970  uspgriedgedg  15971  usgredg2vlem1  16014  usgredg2vlem2  16015