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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 1696 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | 19.29r 1621 | . . 3 ⊢ ((∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑))) | |
3 | hba1 1540 | . . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
4 | pm3.35 347 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑) | |
5 | 3, 4 | exlimih 1593 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ (𝑥 = 𝑦 → ∀𝑥𝜑)) → ∀𝑥𝜑) |
6 | ax-4 1510 | . . . 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 424 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1351 = wceq 1353 ∃wex 1492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-i9 1530 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: equsalh 1726 spimth 1735 spimh 1737 |
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