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Mirrors > Home > ILE Home > Th. List > hbor | GIF version |
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Ref | Expression |
---|---|
hb.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hb.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
hbor | ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hb.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | orc 702 | . . . 4 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
3 | 2 | alimi 1435 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜑 ∨ 𝜓)) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝜑 → ∀𝑥(𝜑 ∨ 𝜓)) |
5 | hb.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
6 | olc 701 | . . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
7 | 6 | alimi 1435 | . . 3 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 ∨ 𝜓)) |
8 | 5, 7 | syl 14 | . 2 ⊢ (𝜓 → ∀𝑥(𝜑 ∨ 𝜓)) |
9 | 4, 8 | jaoi 706 | 1 ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 698 ∀wal 1333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-gen 1429 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: hb3or 1529 nfor 1554 19.43 1608 |
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