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Metamath Proof Explorer

Theorem bnj1241

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1241.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
bnj1241.2 ⊢ ( 𝜓 → 𝐶 = 𝐴 )
Assertion bnj1241 ( ( 𝜑 ∧ 𝜓 ) → 𝐶 ⊆ 𝐵 )

Proof

Step Hyp Ref Expression
1 bnj1241.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2 bnj1241.2 ⊢ ( 𝜓 → 𝐶 = 𝐴 )
3 2 eqcomd ⊢ ( 𝜓 → 𝐴 = 𝐶 )
4 3 adantl ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )
5 1 adantr ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ⊆ 𝐵 )
6 4 5 eqsstrrd ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐶 ⊆ 𝐵 )