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Metamath Proof Explorer

Theorem bj-ax6e

Description: Proof of ax6e (hence ax6 ) from Tarski's system, ax-c9 , ax-c16 . Remark: ax-6 is used only via its principal (unbundled) instance ax6v . (Contributed by BJ, 22-Dec-2020) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion bj-ax6e ∃ 𝑥 𝑥 = 𝑦

Proof

Step Hyp Ref Expression
1 19.2 ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ∃ 𝑥 𝑥 = 𝑦 )
2 1 a1d ⊢ ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∃ 𝑥 𝑥 = 𝑦 ) )
3 bj-ax6elem1 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∀ 𝑥 𝑦 = 𝑧 ) )
4 bj-ax6elem2 ⊢ ( ∀ 𝑥 𝑦 = 𝑧 → ∃ 𝑥 𝑥 = 𝑦 )
5 3 4 syl6 ⊢ ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑦 = 𝑧 → ∃ 𝑥 𝑥 = 𝑦 ) )
6 2 5 pm2.61i ⊢ ( 𝑦 = 𝑧 → ∃ 𝑥 𝑥 = 𝑦 )
7 ax6evr ⊢ ∃ 𝑧 𝑦 = 𝑧
8 6 7 exlimiiv ⊢ ∃ 𝑥 𝑥 = 𝑦