Theorem cbvmo 2241

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
cbvmo	⊢ (∃*xφ ↔ ∃*yψ)

Proof of Theorem cbvmo

Step	Hyp	Ref	Expression
1		cbvmo.1	. . . 4 ⊢ Ⅎyφ
2		cbvmo.2	. . . 4 ⊢ Ⅎxψ
3		cbvmo.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbvex 1985	. . 3 ⊢ (∃xφ ↔ ∃yψ)
5	1, 2, 3	cbveu 2224	. . 3 ⊢ (∃!xφ ↔ ∃!yψ)
6	4, 5	imbi12i 316	. 2 ⊢ ((∃xφ → ∃!xφ) ↔ (∃yψ → ∃!yψ))
7		df-mo 2209	. 2 ⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
8		df-mo 2209	. 2 ⊢ (∃*yψ ↔ (∃yψ → ∃!yψ))
9	6, 7, 8	3bitr4i 268	1 ⊢ (∃*xφ ↔ ∃*yψ)

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544  ∃!weu 2204  ∃*wmo 2205

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  df-mo 2209

This theorem is referenced by:  dffun6f  5124