Theorem cbvexvw 1703

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)

Hypothesis

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

Assertion

Ref	Expression
cbvexvw	⊢ (∃xφ ↔ ∃yψ)

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

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

Proof of Theorem cbvexvw

Step	Hyp	Ref	Expression
1		cbvalvw.1	. . . . 5 ⊢ (x = y → (φ ↔ ψ))
2	1	notbid 285	. . . 4 ⊢ (x = y → (¬ φ ↔ ¬ ψ))
3	2	cbvalvw 1702	. . 3 ⊢ (∀x ¬ φ ↔ ∀y ¬ ψ)
4	3	notbii 287	. 2 ⊢ (¬ ∀x ¬ φ ↔ ¬ ∀y ¬ ψ)
5		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
6		df-ex 1542	. 2 ⊢ (∃yψ ↔ ¬ ∀y ¬ ψ)
7	4, 5, 6	3bitr4i 268	1 ⊢ (∃xφ ↔ ∃yψ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)