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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 1583 | . 2 ⊢ ∃x x = y | |
2 | 19.29r 1509 | . . 3 ⊢ ((∃x x = y ∧ ∀x(x = y → ∀xφ)) → ∃x(x = y ∧ (x = y → ∀xφ))) | |
3 | hba1 1430 | . . . . 5 ⊢ (∀xφ → ∀x∀xφ) | |
4 | pm3.35 329 | . . . . 5 ⊢ ((x = y ∧ (x = y → ∀xφ)) → ∀xφ) | |
5 | 3, 4 | exlimih 1481 | . . . 4 ⊢ (∃x(x = y ∧ (x = y → ∀xφ)) → ∀xφ) |
6 | ax-4 1397 | . . . 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 1240 = wceq 1242 ∃wex 1378 |
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 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: equsalh 1611 spimth 1620 spimh 1622 |
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