Theorem eqimsscd 3987

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 3952	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 3975	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752  df-ss 3915

This theorem is used by:  mhplss  22437  plyconz  26594  precsexlem6  28531  precsexlem7  28532  bdayfinlem  28805  umgr2cycllem  30679  padct  33243  fineqvinfep  35718  nmuladdss  36884  unitscyglem5  43169  mhphf  43547  isubgrvtxuhgr  48884