Theorem cleq1lem 14929

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 4008	. 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 205   ∧ wa 397   = wceq 1542   ⊆ wss 3949

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966

This theorem is referenced by:  cleq1  14930  trcleq12lem  14940  lcmfun  16582  coprmproddvds  16600  isslw  19476  neival  22606  nrmsep3  22859  xkococnlem  23163  ovolval  24990  ovnval2b  45268