Theorem axi11e 2332

Description: Axiom of Variable Substitution for Existence (intuitionistic logic axiom ax-i11e). This can be derived from ax-11 1746 in a classical context but a separate axiom is needed for intuitionistic predicate calculus. (Contributed by Jim Kingdon, 31-Dec-2017.)

Assertion

Ref	Expression
axi11e	⊢ (x = y → (∃x(x = y ∧ φ) → ∃yφ))

Proof of Theorem axi11e

Step	Hyp	Ref	Expression
1		ax-11 1746	. . 3 ⊢ (x = y → (∀y ¬ φ → ∀x(x = y → ¬ φ)))
2		alnex 1543	. . 3 ⊢ (∀y ¬ φ ↔ ¬ ∃yφ)
3		alinexa 1578	. . 3 ⊢ (∀x(x = y → ¬ φ) ↔ ¬ ∃x(x = y ∧ φ))
4	1, 2, 3	3imtr3g 260	. 2 ⊢ (x = y → (¬ ∃yφ → ¬ ∃x(x = y ∧ φ)))
5	4	con4d 97	1 ⊢ (x = y → (∃x(x = y ∧ φ) → ∃yφ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀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-11 1746

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

This theorem is referenced by: (None)