Theorem eubidv 1956

Description: Formula-building rule for unique existential quantifier (deduction form). (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 1466	. 2 ⊢ Ⅎ𝑥𝜑
2		eubidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	eubid 1955	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 103  ∃!weu 1948

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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-eu 1951

This theorem is referenced by:  eubii  1957  eueq2dc  2788  eueq3dc  2789  reuhypd  4291  feu  5187  funfveu  5312  dff4im  5439  acexmid  5643