Theorem eqimsscd 3965

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 3930	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 3953	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3895

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-ss 3896

This theorem is used by:  mhplss  22476  plyconz  26631  precsexlem6  28598  precsexlem7  28599  bdayfinlem  28872  umgr2cycllem  30746  padct  33310  fineqvinfep  35793  nmuladdss  36962  unitscyglem5  43250  mhphf  43626  isubgrvtxuhgr  48962