Theorem eqimssd 4001

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 3967	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 3989	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1568   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-cleq 2762  df-ss 3930

This theorem is referenced by:  eqimss  4003  fssrescdmd  7126  f1ocoima  7305  sraassab  22001  selvvvval  22276  evls1maplmhm  22520  gsumind  33635  fldextrspunlem1  34035  scottsn  35479  r1peuqusdeg1  36093  fourierdlem113  46885  hoicvr  47214  stgrnbgr0  48678