Theorem cleq1lem 14942

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 3947	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
2	1	anbi1d 637	1 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ⊆ wss 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2732  df-ss 3907

This theorem is referenced by:  cleq1  14943  trcleq12lem  14953  lcmfun  16612  coprmproddvds  16630  isslw  19581  neival  23092  nrmsep3  23345  xkococnlem  23649  ovolval  25465  tz9.1regs  35322  ovnval2b  47002