Theorem eqimssd 3964

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 3930	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 3952	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3895

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-ss 3896

This theorem is used by:  eqimss  3966  fssrescdmd  7127  f1ocoima  7311  sraassab  22176  selvvvval  22451  evls1maplmhm  22695  gsumind  33906  fldextrspunlem1  34307  scottsn  35750  r1peuqusdeg1  36408  fourierdlem113  47229  hoicvr  47558  stgrnbgr0  49062