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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 2766 | . . 3 ⊢ (𝜓 → 𝐴 = 𝐶) |
3 | 2 | adantl 473 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
4 | bnj1241.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
5 | 4 | adantr 472 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐵) |
6 | 3, 5 | eqsstr3d 3781 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ⊆ wss 3715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-in 3722 df-ss 3729 |
This theorem is referenced by: bnj1245 31410 bnj1311 31420 |
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