Theorem reu4 3031

Description: Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)

Hypothesis

Ref	Expression
rmo4.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
reu4	⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y)))

Distinct variable groups:   x,y,A   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem reu4

Step	Hyp	Ref	Expression
1		reu5 2825	. 2 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x ∈ A φ))
2		rmo4.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
3	2	rmo4 3030	. . 3 ⊢ (∃*x ∈ A φ ↔ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y))
4	3	anbi2i 675	. 2 ⊢ ((∃x ∈ A φ ∧ ∃*x ∈ A φ) ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y)))
5	1, 4	bitri 240	1 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ ψ) → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617  ∃*wrmo 2618

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by:  reuind  3040  vfinnc  4472  nnpw1ex  4485  ncspw1eu  6160