Theorem ax11inda 1369

Description: Induction step for constructing a substitution instance of ax-11o 1216 without using ax-11o 1216. Quantification case. (When z and y are distinct, ax11inda2 1368 may be used instead to avoid the dummy variable w in the proof.)

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

Proof of Theorem ax11inda

Step	Hyp	Ref	Expression
1		a9e 1123	. . 3 ⊢ ∃w w = y
2		ax11inda.1	. . . . . . 7 ⊢ (¬ ∀x x = w → (x = w → (φ → ∀x(x = w → φ))))
3	2	ax11inda2 1368	. . . . . 6 ⊢ (¬ ∀x x = w → (x = w → (∀zφ → ∀x(x = w → ∀zφ))))
4		dveeq2 1210	. . . . . . . . 9 ⊢ (¬ ∀x x = y → (w = y → ∀x w = y))
5	4	imp 350	. . . . . . . 8 ⊢ ((¬ ∀x x = y ⋀ w = y) → ∀x w = y)
6		hba1 1001	. . . . . . . . . 10 ⊢ (∀x w = y → ∀x∀x w = y)
7		equequ2 1133	. . . . . . . . . . 11 ⊢ (w = y → (x = w ↔ x = y))
8	7	a4s 982	. . . . . . . . . 10 ⊢ (∀x w = y → (x = w ↔ x = y))
9	6, 8	albid 1102	. . . . . . . . 9 ⊢ (∀x w = y → (∀x x = w ↔ ∀x x = y))
10	9	negbid 610	. . . . . . . 8 ⊢ (∀x w = y → (¬ ∀x x = w ↔ ¬ ∀x x = y))
11	5, 10	syl 10	. . . . . . 7 ⊢ ((¬ ∀x x = y ⋀ w = y) → (¬ ∀x x = w ↔ ¬ ∀x x = y))
12	7	adantl 388	. . . . . . . 8 ⊢ ((¬ ∀x x = y ⋀ w = y) → (x = w ↔ x = y))
13	8	imbi1d 612	. . . . . . . . . . 11 ⊢ (∀x w = y → ((x = w → ∀zφ) ↔ (x = y → ∀zφ)))
14	6, 13	albid 1102	. . . . . . . . . 10 ⊢ (∀x w = y → (∀x(x = w → ∀zφ) ↔ ∀x(x = y → ∀zφ)))
15	5, 14	syl 10	. . . . . . . . 9 ⊢ ((¬ ∀x x = y ⋀ w = y) → (∀x(x = w → ∀zφ) ↔ ∀x(x = y → ∀zφ)))
16	15	imbi2d 611	. . . . . . . 8 ⊢ ((¬ ∀x x = y ⋀ w = y) → ((∀zφ → ∀x(x = w → ∀zφ)) ↔ (∀zφ → ∀x(x = y → ∀zφ))))
17	12, 16	imbi12d 625	. . . . . . 7 ⊢ ((¬ ∀x x = y ⋀ w = y) → ((x = w → (∀zφ → ∀x(x = w → ∀zφ))) ↔ (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
18	11, 17	imbi12d 625	. . . . . 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 193	. . . . 5 ⊢ ((¬ ∀x x = y ⋀ w = y) → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
20	19	ex 373	. . . 4 ⊢ (¬ ∀x x = y → (w = y → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))))
21	20	19.23adv 1212	. . 3 ⊢ (¬ ∀x x = y → (∃w w = y → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))))
22	1, 21	mpi 44	. 2 ⊢ (¬ ∀x x = y → (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ)))))
23	22	pm2.43i 64	1 ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954  ∃wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979

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