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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1241 | 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 |
---|---|
bnj1241.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
bnj1241.2 | ⊢ (𝜓 → 𝐶 = 𝐴) |
Ref | Expression |
---|---|
bnj1241 | ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1241.2 | . . . 4 ⊢ (𝜓 → 𝐶 = 𝐴) | |
2 | 1 | eqcomd 2785 | . . 3 ⊢ (𝜓 → 𝐴 = 𝐶) |
3 | 2 | adantl 474 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
4 | bnj1241.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
5 | 4 | adantr 473 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐵) |
6 | 3, 5 | eqsstr3d 3897 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ⊆ wss 3830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-in 3837 df-ss 3844 |
This theorem is referenced by: bnj1245 31928 bnj1311 31938 |
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