Theorem ax11inda 2200

Description: Induction step for constructing a substitution instance of ax-11o 2141 without using ax-11o 2141. Quantification case. (When z and y are distinct, ax11inda2 2199 may be used instead to avoid the dummy variable w in the proof.) (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ax11inda.1	⊢ (¬ ∀x x = w → (x = w → (φ → ∀x(x = w → φ))))

Assertion

Ref	Expression
ax11inda	⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))

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

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem ax11inda

Step	Hyp	Ref	Expression
1		a9ev 1656	. . 3 ⊢ ∃w w = y
2		ax11inda.1	. . . . . . 7 ⊢ (¬ ∀x x = w → (x = w → (φ → ∀x(x = w → φ))))
3	2	ax11inda2 2199	. . . . . 6 ⊢ (¬ ∀x x = w → (x = w → (∀zφ → ∀x(x = w → ∀zφ))))
4		dveeq2-o 2184	. . . . . . . . 9 ⊢ (¬ ∀x x = y → (w = y → ∀x w = y))
5	4	imp 418	. . . . . . . 8 ⊢ ((¬ ∀x x = y ∧ w = y) → ∀x w = y)
6		hba1-o 2149	. . . . . . . . . 10 ⊢ (∀x w = y → ∀x∀x w = y)
7		equequ2 1686	. . . . . . . . . . 11 ⊢ (w = y → (x = w ↔ x = y))
8	7	sps-o 2159	. . . . . . . . . 10 ⊢ (∀x w = y → (x = w ↔ x = y))
9	6, 8	albidh 1590	. . . . . . . . 9 ⊢ (∀x w = y → (∀x x = w ↔ ∀x x = y))
10	9	notbid 285	. . . . . . . 8 ⊢ (∀x w = y → (¬ ∀x x = w ↔ ¬ ∀x x = y))
11	5, 10	syl 15	. . . . . . 7 ⊢ ((¬ ∀x x = y ∧ w = y) → (¬ ∀x x = w ↔ ¬ ∀x x = y))
12	7	adantl 452	. . . . . . . 8 ⊢ ((¬ ∀x x = y ∧ w = y) → (x = w ↔ x = y))
13	8	imbi1d 308	. . . . . . . . . . 11 ⊢ (∀x w = y → ((x = w → ∀zφ) ↔ (x = y → ∀zφ)))
14	6, 13	albidh 1590	. . . . . . . . . 10 ⊢ (∀x w = y → (∀x(x = w → ∀zφ) ↔ ∀x(x = y → ∀zφ)))
15	5, 14	syl 15	. . . . . . . . 9 ⊢ ((¬ ∀x x = y ∧ w = y) → (∀x(x = w → ∀zφ) ↔ ∀x(x = y → ∀zφ)))
16	15	imbi2d 307	. . . . . . . 8 ⊢ ((¬ ∀x x = y ∧ w = y) → ((∀zφ → ∀x(x = w → ∀zφ)) ↔ (∀zφ → ∀x(x = y → ∀zφ))))
17	12, 16	imbi12d 311	. . . . . . 7 ⊢ ((¬ ∀x x = y ∧ w = y) → ((x = w → (∀zφ → ∀x(x = w → ∀zφ))) ↔ (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
18	11, 17	imbi12d 311	. . . . . 6 ⊢ ((¬ ∀x x = y ∧ w = y) → ((¬ ∀x x = w → (x = w → (∀zφ → ∀x(x = w → ∀zφ)))) ↔ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))))
19	3, 18	mpbii 202	. . . . 5 ⊢ ((¬ ∀x x = y ∧ w = y) → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
20	19	ex 423	. . . 4 ⊢ (¬ ∀x x = y → (w = y → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))))
21	20	exlimdv 1636	. . 3 ⊢ (¬ ∀x x = y → (∃w w = y → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))))
22	1, 21	mpi 16	. 2 ⊢ (¬ ∀x x = y → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
23	22	pm2.43i 43	1 ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))

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  ax-16 2144

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)