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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1023 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1023.1 | ⊢ ∃𝑥(𝜑 → 𝜓) |
bnj1023.2 | ⊢ (𝜓 → 𝜒) |
Ref | Expression |
---|---|
bnj1023 | ⊢ ∃𝑥(𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1023.2 | . . . . 5 ⊢ (𝜓 → 𝜒) | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝜑 → 𝜓) → (𝜓 → 𝜒)) |
3 | 2 | ax-gen 1803 | . . 3 ⊢ ∀𝑥((𝜑 → 𝜓) → (𝜓 → 𝜒)) |
4 | bnj1023.1 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) | |
5 | exintr 1900 | . . 3 ⊢ (∀𝑥((𝜑 → 𝜓) → (𝜓 → 𝜒)) → (∃𝑥(𝜑 → 𝜓) → ∃𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜒)))) | |
6 | 3, 4, 5 | mp2 9 | . 2 ⊢ ∃𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜒)) |
7 | pm3.33 765 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜒)) → (𝜑 → 𝜒)) | |
8 | 6, 7 | bnj101 32414 | 1 ⊢ ∃𝑥(𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: bnj1098 32476 bnj1110 32675 bnj1118 32677 bnj1128 32683 bnj1145 32686 |
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