Theorem axc16gALT 2520

Description: Alternate proof of axc16g 2294 that uses df-sb 2090 and requires ax-10 2174, ax-11 2190, ax-13 2402. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc16gALT	⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem axc16gALT

Step	Hyp	Ref	Expression
1		aev 2078	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)
2		axc16ALT 2519	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
3		biidd 264	. . . 4 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝜑 ↔ 𝜑))
4	3	dral1 2469	. . 3 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑧𝜑 ↔ ∀𝑥𝜑))
5	4	biimprd 250	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (∀𝑥𝜑 → ∀𝑧𝜑))
6	1, 2, 5	sylsyld 61	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑))

Syntax hints:   → wi 4  ∀wal 1557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-nf 1803  df-sb 2090

This theorem is referenced by: (None)