Theorem cleq1lem 14693

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 3946	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
2	1	anbi1d 630	1 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  cleq1  14694  trcleq12lem  14704  lcmfun  16350  coprmproddvds  16368  isslw  19213  neival  22253  nrmsep3  22506  xkococnlem  22810  ovolval  24637  ovnval2b  44090