Theorem aevALTOLD 2309

Description: Older alternate proof of aev 1970. Obsolete as of 30-Mar-2021. (Contributed by NM, 8-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
aevALTOLD	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Distinct variable group:   𝑥,𝑦

Proof of Theorem aevALTOLD

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbae 2303	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦)
2		aevlem 1968	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑣)
3		ax7 1930	. . . 4 ⊢ (𝑢 = 𝑤 → (𝑢 = 𝑣 → 𝑤 = 𝑣))
4	3	spimv 2245	. . 3 ⊢ (∀𝑢 𝑢 = 𝑣 → 𝑤 = 𝑣)
5	2, 4	syl 17	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑤 = 𝑣)
6	1, 5	alrimih 1741	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)