Theorem eqimsscd 4039

Description: Equality implies inclusion, deduction version. (Contributed by SN, 15-Feb-2025.)

Hypothesis

Ref	Expression
eqimssd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eqimsscd	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Proof of Theorem eqimsscd

Step	Hyp	Ref	Expression
1		eqimssd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ssid 4004	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 4037	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965

This theorem is referenced by:  precsexlem6  27898  precsexlem7  27899  mhphf  41472