Theorem eubidv 2065

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 1554	. 2 ⊢ Ⅎ𝑥𝜑
2		eubidv.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	eubid 2064	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∃!weu 2057

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 1473  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-4 1536  ax-17 1552  ax-ial 1560

This theorem depends on definitions:  df-bi 117  df-nf 1487  df-eu 2060

This theorem is referenced by:  eubii  2066  eueq2dc  2956  eueq3dc  2957  reuhypd  4539  feu  5484  funfveu  5616  dff4im  5754  acexmid  5973  upxp  14911  dedekindicc  15272