Theorem bieudv 1013

Description: Formula-building rule for uniqueness quantifier (deduction rule).

Hypothesis

Ref	Expression
bieudv.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
bieudv	⊢ (φ → (∃!xψ ↔ ∃!xχ))

Distinct variable group(s):   φ,x

Proof of Theorem bieudv

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (φ → ∀xφ)
2		bieudv.1	. 2 ⊢ (φ → (ψ ↔ χ))
3	1, 2	bieud 1012	1 ⊢ (φ → (∃!xψ ↔ ∃!xχ))

Syntax hints:   → wi 2   ↔ wb 127  ∃!weu 1007

This theorem is referenced by:  euorv 1025  bireudva 1317  eueq2 1429  eueq3 1430  moeq3 1432  reuhyp 1581  fneu 2728  feu 2767  tz6.12-2 2845  fnfvbr 2855  aceq5lem5 3562  aceq5 3563  kmlem2 3581  kmlem11 3590  kmlem12 3591  pjthut 5243

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-eu 1009

metamath.org