Theorem eqimssd 4015

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 3981	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 4003	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1540   ⊆ wss 3926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-ss 3943

This theorem is referenced by:  eqimss  4017  fssrescdmd  7116  f1ocoima  7296  sraassab  21828  evls1maplmhm  22315  fldextrspunlem1  33716  r1peuqusdeg1  35665  selvvvval  42608  stgrnbgr0  47976