Theorem a16g 933

Description: A generalization of axiom ax-16 922.

Assertion

Ref	Expression
a16g	⊢ (∀x x = y → (φ → ∀zφ))

Distinct variable group(s):   x,y

Proof of Theorem a16g

Step	Hyp	Ref	Expression
1		eq5 824	. . 3 ⊢ (∀x x = y → ∀z∀x x = y)
2		ax9a 808	. . . . 5 ⊢ ¬ ∀x ¬ x = z
3		ax-16 922	. . . . 5 ⊢ (∀x x = y → (¬ x = z → ∀x ¬ x = z))
4	2, 3	mt3i 100	. . . 4 ⊢ (∀x x = y → x = z)
5		eqcom 811	. . . 4 ⊢ (x = z → z = x)
6	4, 5	syl 12	. . 3 ⊢ (∀x x = y → z = x)
7	1, 6	19.21ai 740	. 2 ⊢ (∀x x = y → ∀z z = x)
8		ax-16 922	. . 3 ⊢ (∀x x = y → (φ → ∀xφ))
9		idd 11	. . . 4 ⊢ (∀z z = x → (φ → φ))
10	9	del35 836	. . 3 ⊢ (∀z z = x → (∀xφ → ∀zφ))
11	8, 10	syl9r 56	. 2 ⊢ (∀z z = x → (∀x x = y → (φ → ∀zφ)))
12	7, 11	mpcom 49	1 ⊢ (∀x x = y → (φ → ∀zφ))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672   = weq 797

This theorem is referenced by:  a16gb 934

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802  ax-16 922

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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