Theorem ax11v 1705

Description: This is a version of ax-11o 1701 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)

Assertion

Ref	Expression
ax11v	⊢ (x = y → (φ → ∀x(x = y → φ)))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11v

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1583	. 2 ⊢ ∃z z = y
2		ax-17 1416	. . . . 5 ⊢ (φ → ∀zφ)
3		ax-11 1394	. . . . 5 ⊢ (x = z → (∀zφ → ∀x(x = z → φ)))
4	2, 3	syl5 28	. . . 4 ⊢ (x = z → (φ → ∀x(x = z → φ)))
5		equequ2 1596	. . . . 5 ⊢ (z = y → (x = z ↔ x = y))
6	5	imbi1d 220	. . . . . . 7 ⊢ (z = y → ((x = z → φ) ↔ (x = y → φ)))
7	6	albidv 1702	. . . . . 6 ⊢ (z = y → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
8	7	imbi2d 219	. . . . 5 ⊢ (z = y → ((φ → ∀x(x = z → φ)) ↔ (φ → ∀x(x = y → φ))))
9	5, 8	imbi12d 223	. . . 4 ⊢ (z = y → ((x = z → (φ → ∀x(x = z → φ))) ↔ (x = y → (φ → ∀x(x = y → φ)))))
10	4, 9	mpbii 136	. . 3 ⊢ (z = y → (x = y → (φ → ∀x(x = y → φ))))
11	10	exlimiv 1486	. 2 ⊢ (∃z z = y → (x = y → (φ → ∀x(x = y → φ))))
12	1, 11	ax-mp 7	1 ⊢ (x = y → (φ → ∀x(x = y → φ)))

Syntax hints:   → wi 4  ∀wal 1240   = 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-ie2 1380  ax-8 1392  ax-11 1394  ax-17 1416  ax-i9 1420

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  equs5or  1708  sb56  1762