Theorem cbvmow 2603

Description: Rule used to change bound variables, using implicit substitution. Version of cbvmo 2605 with a disjoint variable condition, which does not require ax-10 2137, ax-13 2372. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 23-May-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbvmow	⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvmow

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvmow.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
2		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧
3	1, 2	nfim 1899	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧)
4		cbvmow.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
5		nfv 1917	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧
6	4, 5	nfim 1899	. . . 4 ⊢ Ⅎ𝑥(𝜓 → 𝑦 = 𝑧)
7		cbvmow.3	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8		equequ1 2028	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
9	7, 8	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧)))
10	3, 6, 9	cbvalv1 2338	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧))
11	10	exbii 1850	. 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
12		df-mo 2540	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
13		df-mo 2540	. 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
14	11, 12, 13	3bitr4i 303	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-mo 2540

This theorem is referenced by:  cbveuw  2607  cbvrmow  3375  dffun6f  6448  opabiotafun  6849  2ndcdisj  22607  cbvdisjf  30910  phpreu  35761  mo0sn  46161  isthincd2lem1  46308