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Metamath Proof Explorer
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of
Lemma 7 of KalishMontague p. 86. (Contributed by NM, 19-Apr-2017)
|
|
Ref |
Expression |
|
Hypotheses |
cbvaliw.1 |
⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
|
|
cbvaliw.2 |
⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 ) |
|
|
cbvaliw.3 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) |
|
Assertion |
cbvaliw |
⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cbvaliw.1 |
⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 ) |
| 2 |
|
cbvaliw.2 |
⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 ) |
| 3 |
|
cbvaliw.3 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) ) |
| 4 |
2 3
|
spimw |
⊢ ( ∀ 𝑥 𝜑 → 𝜓 ) |
| 5 |
1 4
|
alrimih |
⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 ) |