Theorem eubidv 2212

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 1619	. 2 ⊢ Ⅎxφ
2		eubidv.1	. 2 ⊢ (φ → (ψ ↔ χ))
3	1, 2	eubid 2211	1 ⊢ (φ → (∃!xψ ↔ ∃!xχ))

Syntax hints:   → wi 4   ↔ wb 176  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  eubii  2213  eueq2  3010  eueq3  3011  moeq3  3013  fneu  5187  feu  5242  dff4  5421  scancan  6331