Theorem ax11ev 1706

Description: Analogue to ax11v 1705 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)

Assertion

Ref	Expression
ax11ev	⊢ (x = y → (∃x(x = y ∧ φ) → φ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11ev

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1583	. 2 ⊢ ∃z z = y
2		ax11e 1674	. . . . 5 ⊢ (x = z → (∃x(x = z ∧ φ) → ∃zφ))
3		ax-17 1416	. . . . . 6 ⊢ (φ → ∀zφ)
4	3	19.9h 1531	. . . . 5 ⊢ (∃zφ ↔ φ)
5	2, 4	syl6ib 150	. . . 4 ⊢ (x = z → (∃x(x = z ∧ φ) → φ))
6		equequ2 1596	. . . . 5 ⊢ (z = y → (x = z ↔ x = y))
7	6	anbi1d 438	. . . . . . 7 ⊢ (z = y → ((x = z ∧ φ) ↔ (x = y ∧ φ)))
8	7	exbidv 1703	. . . . . 6 ⊢ (z = y → (∃x(x = z ∧ φ) ↔ ∃x(x = y ∧ φ)))
9	8	imbi1d 220	. . . . 5 ⊢ (z = y → ((∃x(x = z ∧ φ) → φ) ↔ (∃x(x = y ∧ φ) → φ)))
10	6, 9	imbi12d 223	. . . 4 ⊢ (z = y → ((x = z → (∃x(x = z ∧ φ) → φ)) ↔ (x = y → (∃x(x = y ∧ φ) → φ))))
11	5, 10	mpbii 136	. . 3 ⊢ (z = y → (x = y → (∃x(x = y ∧ φ) → φ)))
12	11	exlimiv 1486	. 2 ⊢ (∃z z = y → (x = y → (∃x(x = y ∧ φ) → φ)))
13	1, 12	ax-mp 7	1 ⊢ (x = y → (∃x(x = y ∧ φ) → φ))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)