Theorem eqimssd 3986

Description: Equality implies inclusion, deduction version. (Contributed by SN, 6-Nov-2024.)

Hypothesis

Ref	Expression
eqimssd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eqimssd	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem eqimssd

Step	Hyp	Ref	Expression
1		eqimssd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		ssid 3952	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 3974	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-ss 3914

This theorem is referenced by:  eqimss  3988  fssrescdmd  7059  f1ocoima  7237  sraassab  21805  evls1maplmhm  22292  gsumind  33310  fldextrspunlem1  33688  r1peuqusdeg1  35687  selvvvval  42677  stgrnbgr0  48063