Theorem eqimssd 3973

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 3939	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 3961	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1548   ⊆ wss 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-cleq 2733  df-ss 3902

This theorem is referenced by:  eqimss  3975  fssrescdmd  7072  f1ocoima  7251  sraassab  21847  selvvvval  22122  evls1maplmhm  22367  gsumind  33432  fldextrspunlem1  33871  r1peuqusdeg1  35886  hoicvr  47005  stgrnbgr0  48469