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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1101 | 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 |
---|---|
bnj1101.1 | ⊢ ∃𝑥(𝜑 → 𝜓) |
bnj1101.2 | ⊢ (𝜒 → 𝜑) |
Ref | Expression |
---|---|
bnj1101 | ⊢ ∃𝑥(𝜒 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1101.1 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) | |
2 | pm3.42 494 | . . 3 ⊢ ((𝜑 → 𝜓) → ((𝜒 ∧ 𝜑) → 𝜓)) | |
3 | 1, 2 | bnj101 32688 | . 2 ⊢ ∃𝑥((𝜒 ∧ 𝜑) → 𝜓) |
4 | bnj1101.2 | . . . . 5 ⊢ (𝜒 → 𝜑) | |
5 | 4 | pm4.71i 560 | . . . 4 ⊢ (𝜒 ↔ (𝜒 ∧ 𝜑)) |
6 | 5 | imbi1i 350 | . . 3 ⊢ ((𝜒 → 𝜓) ↔ ((𝜒 ∧ 𝜑) → 𝜓)) |
7 | 6 | exbii 1850 | . 2 ⊢ (∃𝑥(𝜒 → 𝜓) ↔ ∃𝑥((𝜒 ∧ 𝜑) → 𝜓)) |
8 | 3, 7 | mpbir 230 | 1 ⊢ ∃𝑥(𝜒 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 |
This theorem is referenced by: bnj1110 32948 bnj1128 32956 bnj1145 32959 |
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