Theorem eqimssd 3990

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 3956	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 3978	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-ss 3919

This theorem is used by:  eqimss  3992  fssrescdmd  7123  f1ocoima  7307  sraassab  22084  selvvvval  22359  evls1maplmhm  22603  gsumind  33772  fldextrspunlem1  34172  scottsn  35620  r1peuqusdeg1  36209  fourierdlem113  47034  hoicvr  47363  stgrnbgr0  48867