Theorem cleq1lem 14891

Description: Equality implies bijection. (Contributed by RP, 9-May-2020.)

Assertion

Ref	Expression
cleq1lem	⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑)))

Proof of Theorem cleq1lem

Step	Hyp	Ref	Expression
1		sseq1 3956	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
2	1	anbi1d 631	1 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ⊆ wss 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2725  df-ss 3915

This theorem is referenced by:  cleq1  14892  trcleq12lem  14902  lcmfun  16558  coprmproddvds  16576  isslw  19522  neival  23018  nrmsep3  23271  xkococnlem  23575  ovolval  25402  tz9.1regs  35151  ovnval2b  46674