Theorem cbvexdva 2011

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
cbvaldva.1	⊢ ((φ ∧ x = y) → (ψ ↔ χ))

Assertion

Ref	Expression
cbvexdva	⊢ (φ → (∃xψ ↔ ∃yχ))

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

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

Proof of Theorem cbvexdva

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎyφ
2		nfvd 1620	. 2 ⊢ (φ → Ⅎyψ)
3		cbvaldva.1	. . 3 ⊢ ((φ ∧ x = y) → (ψ ↔ χ))
4	3	ex 423	. 2 ⊢ (φ → (x = y → (ψ ↔ χ)))
5	1, 2, 4	cbvexd 2009	1 ⊢ (φ → (∃xψ ↔ ∃yχ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃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  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbvrexdva2  2841