Theorem cbveu 2224

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
cbveu.1	⊢ Ⅎyφ
cbveu.2	⊢ Ⅎxψ
cbveu.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbveu	⊢ (∃!xφ ↔ ∃!yψ)

Proof of Theorem cbveu

Step	Hyp	Ref	Expression
1		cbveu.1	. . 3 ⊢ Ⅎyφ
2	1	sb8eu 2222	. 2 ⊢ (∃!xφ ↔ ∃!y[y / x]φ)
3		cbveu.2	. . . 4 ⊢ Ⅎxψ
4		cbveu.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
5	3, 4	sbie 2038	. . 3 ⊢ ([y / x]φ ↔ ψ)
6	5	eubii 2213	. 2 ⊢ (∃!y[y / x]φ ↔ ∃!yψ)
7	2, 6	bitri 240	1 ⊢ (∃!xφ ↔ ∃!yψ)

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  [wsb 1648  ∃!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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

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

This theorem is referenced by:  cbvmo  2241  cbvreu  2834  cbvreucsf  3201  tz6.12-1  5345