Theorem eqimsscd 3993

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 3958	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 3981	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-ss 3921

This theorem is used by:  mhplss  22329  precsexlem6  28416  precsexlem7  28417  bdayfinlem  28690  padct  33074  fineqvinfep  35546  nmuladdss  36713  unitscyglem5  42994  mhphf  43357  isubgrvtxuhgr  48657