Theorem ax11vALT 2097

Description: Alternate proof of ax11v 2096 that avoids Theorem ax16 2045 and is proved directly from ax-11 1746 rather than via ax11o 1994. (Contributed by Jim Kingdon, 15-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ax11vALT	⊢ (x = y → (φ → ∀x(x = y → φ)))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11vALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1951	. 2 ⊢ ∃z z = y
2		ax-17 1616	. . . . 5 ⊢ (φ → ∀zφ)
3		ax-11 1746	. . . . 5 ⊢ (x = z → (∀zφ → ∀x(x = z → φ)))
4	2, 3	syl5 28	. . . 4 ⊢ (x = z → (φ → ∀x(x = z → φ)))
5		equequ2 1686	. . . . 5 ⊢ (z = y → (x = z ↔ x = y))
6	5	imbi1d 308	. . . . . . 7 ⊢ (z = y → ((x = z → φ) ↔ (x = y → φ)))
7	6	albidv 1625	. . . . . 6 ⊢ (z = y → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
8	7	imbi2d 307	. . . . 5 ⊢ (z = y → ((φ → ∀x(x = z → φ)) ↔ (φ → ∀x(x = y → φ))))
9	5, 8	imbi12d 311	. . . 4 ⊢ (z = y → ((x = z → (φ → ∀x(x = z → φ))) ↔ (x = y → (φ → ∀x(x = y → φ)))))
10	4, 9	mpbii 202	. . 3 ⊢ (z = y → (x = y → (φ → ∀x(x = y → φ))))
11	10	exlimiv 1634	. 2 ⊢ (∃z z = y → (x = y → (φ → ∀x(x = y → φ))))
12	1, 11	ax-mp 5	1 ⊢ (x = y → (φ → ∀x(x = y → φ)))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541   = wceq 1642

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: (None)