Theorem cbvmo 2506

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 2272	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
5	1, 2, 3	cbveu 2505	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
6	4, 5	imbi12i 340	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
7		df-mo 2475	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8		df-mo 2475	. 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
9	6, 7, 8	3bitr4i 292	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∃wex 1704  Ⅎwnf 1708  ∃!weu 2470  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475

This theorem is referenced by:  dffun6f  5902  opabiotafun  6259  2ndcdisj  21259  cbvdisjf  29385  phpreu  33393