Theorem a16g 1864

Description: A generalization of Axiom ax-16 1814. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem a16g

Step	Hyp	Ref	Expression
1		aev 1812	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2		ax16 1813	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3		biidd 172	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝜑 ↔ 𝜑))
4	3	dral1 1730	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
5	4	biimprd 158	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑))
6	1, 2, 5	sylsyld 58	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Syntax hints:   → wi 4  ∀wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  a16gb  1865  a16nf  1866