Theorem bimod 1030

Description: Formula-building rule for "at most one" quantifier (deduction rule).

Hypotheses

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

Assertion

Ref	Expression
bimod	⊢ (φ → (∃*xψ ↔ ∃*xχ))

Proof of Theorem bimod

Step	Hyp	Ref	Expression
1		bimod.1	. . . 4 ⊢ (φ → ∀xφ)
2		bimod.2	. . . 4 ⊢ (φ → (ψ ↔ χ))
3	1, 2	biexd 783	. . 3 ⊢ (φ → (∃xψ ↔ ∃xχ))
4	1, 2	bieud 1012	. . 3 ⊢ (φ → (∃!xψ ↔ ∃!xχ))
5	3, 4	imbi12d 474	. 2 ⊢ (φ → ((∃xψ → ∃!xψ) ↔ (∃xχ → ∃!xχ)))
6		df-mo 1010	. 2 ⊢ (∃*xψ ↔ (∃xψ → ∃!xψ))
7		df-mo 1010	. 2 ⊢ (∃*xχ ↔ (∃xχ → ∃!xχ))
8	5, 6, 7	3bitr4g 428	1 ⊢ (φ → (∃*xψ ↔ ∃*xχ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678  ∃!weu 1007  ∃*wmo 1008

This theorem is referenced by:  bimo 1031  mosubop 1911  dffunmof 2678

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  df-mo 1010

metamath.org