Theorem eqimssd 3991

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 3957	. 2 ⊢ 𝐵 ⊆ 𝐵
3	1, 2	eqsstrdi 3979	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ss 3919

This theorem is referenced by:  eqimss  3993  fssrescdmd  7073  f1ocoima  7251  sraassab  21827  evls1maplmhm  22325  gsumind  33428  fldextrspunlem1  33834  r1peuqusdeg1  35839  selvvvval  42895  stgrnbgr0  48277