Theorem eqimsscd 4002

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 3967	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 3990	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1568   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-cleq 2762  df-ss 3930

This theorem is referenced by:  mhplss  22301  precsexlem6  28385  precsexlem7  28386  bdayfinlem  28659  padct  33033  fineqvinfep  35496  unitscyglem5  42916  mhphf  43281  isubgrvtxuhgr  48578