Theorem eubidv 2021

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 1515	. 2 ⊢ Ⅎ𝑥𝜑
2		eubidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	eubid 2020	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∃!weu 2013

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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-eu 2016

This theorem is referenced by:  eubii  2022  eueq2dc  2894  eueq3dc  2895  reuhypd  4443  feu  5364  funfveu  5493  dff4im  5625  acexmid  5835  upxp  12813  dedekindicc  13152