Theorem aevlem 1968

Description: Lemma for aev 1970 and axc16g 2119. Change free and bound variables. Instance of aev 1970. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2234, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)

Assertion

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

Distinct variable groups:   𝑥,𝑦   𝑧,𝑡

Proof of Theorem aevlem

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvaev 1966	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑦)
2		aevlem0 1967	. 2 ⊢ (∀𝑢 𝑢 = 𝑦 → ∀𝑥 𝑥 = 𝑢)
3		cbvaev 1966	. 2 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑡 𝑡 = 𝑢)
4		aevlem0 1967	. 2 ⊢ (∀𝑡 𝑡 = 𝑢 → ∀𝑧 𝑧 = 𝑡)
5	1, 2, 3, 4	4syl 19	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)

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

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  aeveq  1969  aev  1970  hbaevg  1971  axc16g  2119  axc11vOLD  2126  axc16gOLD  2147  aevOLD  2148  aevALTOLD  2309  bj-axc16g16  31861  bj-axc11nv  31933