Description: Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995)
Ref | Expression | ||
---|---|---|---|
Assertion | excom13 | ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑥 𝜑 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom | ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 ∃ 𝑧 𝜑 ) | |
2 | excom | ⊢ ( ∃ 𝑥 ∃ 𝑧 𝜑 ↔ ∃ 𝑧 ∃ 𝑥 𝜑 ) | |
3 | 2 | exbii | ⊢ ( ∃ 𝑦 ∃ 𝑥 ∃ 𝑧 𝜑 ↔ ∃ 𝑦 ∃ 𝑧 ∃ 𝑥 𝜑 ) |
4 | excom | ⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑥 𝜑 ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑥 𝜑 ) | |
5 | 1 3 4 | 3bitri | ⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑥 𝜑 ) |