Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of KalishMontague p. 86. (Contributed by Wolf Lammen, 5-Mar-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | wl-cbvmotv | ⊢ ( ∃* 𝑥 ⊤ → ∃* 𝑦 ⊤ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax7v2 | ⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝑧 → 𝑦 = 𝑧 ) ) | |
| 2 | 1 | imim2d | ⊢ ( 𝑥 = 𝑦 → ( ( ⊤ → 𝑥 = 𝑧 ) → ( ⊤ → 𝑦 = 𝑧 ) ) ) |
| 3 | 2 | cbvalivw | ⊢ ( ∀ 𝑥 ( ⊤ → 𝑥 = 𝑧 ) → ∀ 𝑦 ( ⊤ → 𝑦 = 𝑧 ) ) |
| 4 | 3 | eximi | ⊢ ( ∃ 𝑧 ∀ 𝑥 ( ⊤ → 𝑥 = 𝑧 ) → ∃ 𝑧 ∀ 𝑦 ( ⊤ → 𝑦 = 𝑧 ) ) |
| 5 | df-mo | ⊢ ( ∃* 𝑥 ⊤ ↔ ∃ 𝑧 ∀ 𝑥 ( ⊤ → 𝑥 = 𝑧 ) ) | |
| 6 | df-mo | ⊢ ( ∃* 𝑦 ⊤ ↔ ∃ 𝑧 ∀ 𝑦 ( ⊤ → 𝑦 = 𝑧 ) ) | |
| 7 | 4 5 6 | 3imtr4i | ⊢ ( ∃* 𝑥 ⊤ → ∃* 𝑦 ⊤ ) |