Theorem eubidv 1949

Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Hypothesis

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

Assertion

Ref	Expression
eubidv	⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem eubidv

Step	Hyp	Ref	Expression
1		nfv 1461	. 2 ⊢ Ⅎ𝑥𝜑
2		eubidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	eubid 1948	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 103  ∃!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-eu 1944

This theorem is referenced by:  eubii  1950  eueq2dc  2765  eueq3dc  2766  reuhypd  4221  feu  5092  funfveu  5208  dff4im  5334  acexmid  5531