Theorem eqimsscd 3995

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 3960	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 3983	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1562   ⊆ wss 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-cleq 2756  df-ss 3923

This theorem is referenced by:  mhplss  22222  precsexlem6  28307  precsexlem7  28308  bdayfinlem  28581  padct  32922  fineqvinfep  35425  unitscyglem5  42821  mhphf  43184  isubgrvtxuhgr  48491