Theorem bj-ax6elem2 37346

Description: Lemma for bj-ax6e 37347. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax6elem2	⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑧

Proof of Theorem bj-ax6elem2

Step	Hyp	Ref	Expression
1		ax6ev 2002	. . 3 ⊢ ∃𝑥 𝑥 = 𝑧
2		equeucl 2057	. . 3 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))
3	1, 2	eximii 1870	. 2 ⊢ ∃𝑥(𝑦 = 𝑧 → 𝑥 = 𝑦)
4	3	19.35i 1911	1 ⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  bj-ax6e  37347