Theorem ax16 1785

Description: Theorem showing that ax-16 1786 is redundant if ax-17 1506 is included in the axiom system. The important part of the proof is provided by aev 1784.

See ax16ALT 1831 for an alternate proof that does not require ax-10 1483 or ax-12 1489.

This theorem should not be referenced in any proof. Instead, use ax-16 1786 below so that theorems needing ax-16 1786 can be more easily identified. (Contributed by NM, 8-Nov-2006.)

Assertion

Ref	Expression
ax16	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax16

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1784	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑧)
2		ax-17 1506	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
3		sbequ12 1744	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	3	biimpcd 158	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑥]𝜑))
5	2, 4	alimdh 1443	. . 3 ⊢ (𝜑 → (∀𝑧 𝑥 = 𝑧 → ∀𝑧[𝑧 / 𝑥]𝜑))
6	2	hbsb3 1780	. . . 4 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑥[𝑧 / 𝑥]𝜑)
7		stdpc7 1743	. . . 4 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜑 → 𝜑))
8	6, 2, 7	cbv3h 1721	. . 3 ⊢ (∀𝑧[𝑧 / 𝑥]𝜑 → ∀𝑥𝜑)
9	5, 8	syl6com 35	. 2 ⊢ (∀𝑧 𝑥 = 𝑧 → (𝜑 → ∀𝑥𝜑))
10	1, 9	syl 14	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1329  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  dveeq2  1787  dveeq2or  1788  a16g  1836  exists2  2096