Theorem eqimsscd 3992

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 3957	. 2 ⊢ 𝐴 ⊆ 𝐴
3	1, 2	eqsstrrdi 3980	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3902

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ss 3919

This theorem is referenced by:  mhplss  22102  precsexlem6  28212  precsexlem7  28213  bdayfinlem  28486  fineqvinfep  35283  unitscyglem5  42521  mhphf  42907  isubgrvtxuhgr  48177