Theorem cbvmovw 2628

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo 2630 and cbvmow 2629 for versions with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995.) (Revised by GG, 30-Sep-2024.)

Hypothesis

Ref	Expression
cbvmovw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvmovw	⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

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

Proof of Theorem cbvmovw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvmovw.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2		equequ1 2044	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
3	1, 2	imbi12d 346	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧)))
4	3	cbvalvw 2055	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧))
5	4	exbii 1867	. 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
6		dfmo 2566	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
7		dfmo 2566	. 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧))
8	5, 6, 7	3bitr4i 305	1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557  ∃wex 1798  ∃*wmo 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-mo 2565

This theorem is referenced by:  cbveuvw  2631  cbvrmovw  3387  cbvrmovw2  36549