Theorem eubidv 2094

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 1581	. 2 ⊢ Ⅎ𝑥𝜑
2		eubidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	eubid 2093	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃!weu 2086

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-eu 2089

This theorem is referenced by:  eubii  2095  eueq2dc  2999  eueq3dc  3000  reuhypd  4615  feu  5572  funfveu  5706  dff4im  5848  acexmid  6078  upxp  15356  dedekindicc  15717