Theorem eqimsscd 3990

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 3955	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 3978	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2727  df-ss 3917

This theorem is referenced by:  mhplss  22100  precsexlem6  28191  precsexlem7  28192  bdayfinlem  28463  fineqvinfep  35260  unitscyglem5  42488  mhphf  42877  isubgrvtxuhgr  48147