Theorem cbvmo 2117

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 1802	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
5	1, 2, 3	cbveu 2101	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
6	4, 5	imbi12i 239	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
7		df-mo 2081	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8		df-mo 2081	. 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
9	6, 7, 8	3bitr4i 212	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1506  ∃wex 1538  ∃!weu 2077  ∃*wmo 2078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  cbvmow  2118  dffun6f  5330