Theorem aeveq 2060

Description: The antecedent ∀𝑥𝑥 = 𝑦 with a disjoint variable condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.)

Assertion

Ref	Expression
aeveq	⊢ (∀𝑥 𝑥 = 𝑦 → 𝑧 = 𝑡)

Distinct variable group:   𝑥,𝑦

Proof of Theorem aeveq

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 2059	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑧)
2		ax6ev 1971	. . 3 ⊢ ∃𝑢 𝑢 = 𝑡
3		ax7 2018	. . . 4 ⊢ (𝑢 = 𝑧 → (𝑢 = 𝑡 → 𝑧 = 𝑡))
4	3	aleximi 1834	. . 3 ⊢ (∀𝑢 𝑢 = 𝑧 → (∃𝑢 𝑢 = 𝑡 → ∃𝑢 𝑧 = 𝑡))
5	2, 4	mpi 20	. 2 ⊢ (∀𝑢 𝑢 = 𝑧 → ∃𝑢 𝑧 = 𝑡)
6		ax5e 1914	. 2 ⊢ (∃𝑢 𝑧 = 𝑡 → 𝑧 = 𝑡)
7	1, 5, 6	3syl 18	1 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑧 = 𝑡)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  aev  2061  2ax6e  2476  aevdemo  30547  wl-moteq  37766  wl-spae  37773