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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 2804 | . . 3 ⊢ (𝜓 → 𝐴 = 𝐶) |
3 | 2 | adantl 485 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
4 | bnj1241.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
5 | 4 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐵) |
6 | 3, 5 | eqsstrrd 3954 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ⊆ wss 3881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 |
This theorem is referenced by: bnj1245 32396 bnj1311 32406 |
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