Theorem eqimssd 3992

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 3958	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 3980	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-cleq 2754  df-ss 3921

This theorem is used by:  eqimss  3994  fssrescdmd  7122  f1ocoima  7301  sraassab  22029  selvvvval  22304  evls1maplmhm  22548  gsumind  33674  fldextrspunlem1  34074  scottsn  35528  r1peuqusdeg1  36143  fourierdlem113  46961  hoicvr  47290  stgrnbgr0  48757