Theorem bj-ax6elem2 36650

Description: Lemma for bj-ax6e 36651. (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 1967	. . 3 ⊢ ∃𝑥 𝑥 = 𝑧
2		equeucl 2021	. . 3 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))
3	1, 2	eximii 1834	. 2 ⊢ ∃𝑥(𝑦 = 𝑧 → 𝑥 = 𝑦)
4	3	19.35i 1876	1 ⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777

This theorem is referenced by:  bj-ax6e  36651