Theorem ax11v2-o 2201

Description: Recovery of ax-11o 2141 from ax11v 2096 without using ax-11o 2141. The hypothesis is even weaker than ax11v 2096, with z both distinct from x and not occurring in φ. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11o 2141. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax11v2-o.1	⊢ (x = z → (φ → ∀x(x = z → φ)))

Assertion

Ref	Expression
ax11v2-o	⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Distinct variable groups:   x,z   y,z   φ,z

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11v2-o

Step	Hyp	Ref	Expression
1		a9ev 1656	. 2 ⊢ ∃z z = y
2		ax11v2-o.1	. . . . 5 ⊢ (x = z → (φ → ∀x(x = z → φ)))
3		equequ2 1686	. . . . . . 7 ⊢ (z = y → (x = z ↔ x = y))
4	3	adantl 452	. . . . . 6 ⊢ ((¬ ∀x x = y ∧ z = y) → (x = z ↔ x = y))
5		dveeq2-o 2184	. . . . . . . . 9 ⊢ (¬ ∀x x = y → (z = y → ∀x z = y))
6	5	imp 418	. . . . . . . 8 ⊢ ((¬ ∀x x = y ∧ z = y) → ∀x z = y)
7		nfa1-o 2166	. . . . . . . . 9 ⊢ Ⅎx∀x z = y
8	3	imbi1d 308	. . . . . . . . . 10 ⊢ (z = y → ((x = z → φ) ↔ (x = y → φ)))
9	8	sps-o 2159	. . . . . . . . 9 ⊢ (∀x z = y → ((x = z → φ) ↔ (x = y → φ)))
10	7, 9	albid 1772	. . . . . . . 8 ⊢ (∀x z = y → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
11	6, 10	syl 15	. . . . . . 7 ⊢ ((¬ ∀x x = y ∧ z = y) → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
12	11	imbi2d 307	. . . . . 6 ⊢ ((¬ ∀x x = y ∧ z = y) → ((φ → ∀x(x = z → φ)) ↔ (φ → ∀x(x = y → φ))))
13	4, 12	imbi12d 311	. . . . 5 ⊢ ((¬ ∀x x = y ∧ z = y) → ((x = z → (φ → ∀x(x = z → φ))) ↔ (x = y → (φ → ∀x(x = y → φ)))))
14	2, 13	mpbii 202	. . . 4 ⊢ ((¬ ∀x x = y ∧ z = y) → (x = y → (φ → ∀x(x = y → φ))))
15	14	ex 423	. . 3 ⊢ (¬ ∀x x = y → (z = y → (x = y → (φ → ∀x(x = y → φ)))))
16	15	exlimdv 1636	. 2 ⊢ (¬ ∀x x = y → (∃z z = y → (x = y → (φ → ∀x(x = y → φ)))))
17	1, 16	mpi 16	1 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ 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-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

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:  ax11a2-o  2202