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Mirrors > Home > ILE Home > Th. List > ax9o | GIF version |
Description: An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Ref | Expression |
---|---|
ax9o | ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a9e 1586 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | 19.29r 1512 | . . 3 ⊢ ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑))) | |
3 | hba1 1433 | . . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
4 | pm3.35 329 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑) | |
5 | 3, 4 | exlimih 1484 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑) |
6 | ax-4 1400 | . . . 4 ⊢ (∀𝑥𝜑 → 𝜑) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → 𝜑) |
8 | 2, 7 | syl 14 | . 2 ⊢ ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → 𝜑) |
9 | 1, 8 | mpan 400 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: equsalh 1614 spimth 1623 spimh 1625 |
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