Theorem bieud 1012

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

Hypotheses

Ref	Expression
bieud.1	⊢ (φ → ∀xφ)
bieud.2	⊢ (φ → (ψ ↔ χ))

Assertion

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

Proof of Theorem bieud

Step	Hyp	Ref	Expression
1		bieud.1	. . . 4 ⊢ (φ → ∀xφ)
2		bieud.2	. . . . 5 ⊢ (φ → (ψ ↔ χ))
3	2	bibi1d 471	. . . 4 ⊢ (φ → ((ψ ↔ x = y) ↔ (χ ↔ x = y)))
4	1, 3	biald 782	. . 3 ⊢ (φ → (∀x(ψ ↔ x = y) ↔ ∀x(χ ↔ x = y)))
5	4	biexdv 936	. 2 ⊢ (φ → (∃y∀x(ψ ↔ x = y) ↔ ∃y∀x(χ ↔ x = y)))
6		df-eu 1009	. 2 ⊢ (∃!xψ ↔ ∃y∀x(ψ ↔ x = y))
7		df-eu 1009	. 2 ⊢ (∃!xχ ↔ ∃y∀x(χ ↔ x = y))
8	5, 6, 7	3bitr4g 428	1 ⊢ (φ → (∃!xψ ↔ ∃!xχ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007

This theorem is referenced by:  bieudv 1013  bieu 1014  bimod 1030  reueqf 1323

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

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