Theorem cleq1lem 13569

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 3589	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶))
2	1	anbi1d 737	1 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554

This theorem is referenced by:  cleq1  13570  trcleq12lem  13580  lcmfun  15196  coprmproddvds  15215  ovnval2b  39442