Theorem ax16ALT2 2048

Description: Alternate proof of ax16 2045. (Contributed by NM, 8-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax16ALT2	⊢ (∀x x = y → (φ → ∀xφ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem ax16ALT2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1991	. 2 ⊢ (∀x x = y → ∀z x = z)
2		sbequ12 1919	. . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ))
3	2	biimpcd 215	. . . 4 ⊢ (φ → (x = z → [z / x]φ))
4	3	alimdv 1621	. . 3 ⊢ (φ → (∀z x = z → ∀z[z / x]φ))
5		nfv 1619	. . . . 5 ⊢ Ⅎzφ
6	5	nfs1 2044	. . . 4 ⊢ Ⅎx[z / x]φ
7		stdpc7 1917	. . . 4 ⊢ (z = x → ([z / x]φ → φ))
8	6, 5, 7	cbv3 1982	. . 3 ⊢ (∀z[z / x]φ → ∀xφ)
9	4, 8	syl6com 31	. 2 ⊢ (∀z x = z → (φ → ∀xφ))
10	1, 9	syl 15	1 ⊢ (∀x x = y → (φ → ∀xφ))

Syntax hints:   → wi 4  ∀wal 1540  [wsb 1648

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  df-sb 1649

This theorem is referenced by:  a16gALT  2049