Theorem reubidv 2649

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 17-Oct-1996.)

Hypothesis

Ref	Expression
reubidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem reubidv

Step	Hyp	Ref	Expression
1		reubidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	adantr 274	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
3	2	reubidva 2648	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2136  ∃!wreu 2446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-eu 2017  df-reu 2451

This theorem is referenced by:  reueqd  2671  sbcreug  3031  xpf1o  6810  srpospr  7724  creur  8854  creui  8855  divalg2  11863