Theorem aev 1983

Description: A "distinctor elimination" lemma with no restrictions on variables in the consequent. (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.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 19-Mar-2021.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem aev

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aevlem 1981	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑣 𝑣 = 𝑤)
2		aeveq 1982	. . 3 ⊢ (∀𝑣 𝑣 = 𝑤 → 𝑡 = 𝑢)
3	2	alrimiv 1855	. 2 ⊢ (∀𝑣 𝑣 = 𝑤 → ∀𝑧 𝑡 = 𝑢)
4	1, 3	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢)

Syntax hints:   → wi 4  ∀wal 1481

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

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

This theorem is referenced by:  aev2  1986  aev2ALT  1987  axc16nfOLD  2163  axc11n  2307  axc11nOLD  2308  axc16gALT  2367  aevdemo  27317  axc11n11r  32673  wl-naev  33302  wl-hbnaev  33305  wl-ax11-lem2  33363