Description: Obsolete version of 2ax6e as of 3-Oct-2023. (Contributed by Wolf Lammen, 28-Sep-2018) (Proof modification is discouraged.) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | 2ax6eOLD | ⊢ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aeveq | ⊢ ( ∀ 𝑤 𝑤 = 𝑧 → 𝑧 = 𝑥 ) | |
2 | aeveq | ⊢ ( ∀ 𝑤 𝑤 = 𝑧 → 𝑤 = 𝑦 ) | |
3 | 1 2 | jca | ⊢ ( ∀ 𝑤 𝑤 = 𝑧 → ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) |
4 | 19.8a | ⊢ ( ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) | |
5 | 19.8a | ⊢ ( ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) | |
6 | 3 4 5 | 3syl | ⊢ ( ∀ 𝑤 𝑤 = 𝑧 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) |
7 | 2ax6elem | ⊢ ( ¬ ∀ 𝑤 𝑤 = 𝑧 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) ) | |
8 | 6 7 | pm2.61i | ⊢ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) |