Theorem aevOLD 2162

Description: Obsolete proof of aev 1983 as of 19-Mar-2021. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2246, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem aevOLD

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 1981	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑤)
2		ax6ev 1890	. . . 4 ⊢ ∃𝑢 𝑢 = 𝑣
3		ax7 1943	. . . . 5 ⊢ (𝑢 = 𝑤 → (𝑢 = 𝑣 → 𝑤 = 𝑣))
4	3	aleximi 1759	. . . 4 ⊢ (∀𝑢 𝑢 = 𝑤 → (∃𝑢 𝑢 = 𝑣 → ∃𝑢 𝑤 = 𝑣))
5	2, 4	mpi 20	. . 3 ⊢ (∀𝑢 𝑢 = 𝑤 → ∃𝑢 𝑤 = 𝑣)
6		ax5e 1841	. . 3 ⊢ (∃𝑢 𝑤 = 𝑣 → 𝑤 = 𝑣)
7	1, 5, 6	3syl 18	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑤 = 𝑣)
8		axc16g 2134	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = 𝑣 → ∀𝑧 𝑤 = 𝑣))
9	7, 8	mpd 15	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣)

Syntax hints:   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)