Theorem eqelb 36382

Description: Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 17-Jul-2019.)

Assertion

Ref	Expression
eqelb	⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))

Proof of Theorem eqelb

Step	Hyp	Ref	Expression
1		simpl 483	. . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 = 𝐴)
2		eqeltr 36381	. . . 4 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐵 ∈ 𝐶)
3	1, 2	jca 512	. . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) → (𝐵 = 𝐴 ∧ 𝐵 ∈ 𝐶))
4		eqcom 2745	. . . 4 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
5	4	anbi1i 624	. . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶))
6	4	anbi1i 624	. . 3 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))
7	3, 5, 6	3imtr3i 291	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) → (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))
8		simpl 483	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 = 𝐵)
9		eqeltr 36381	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶)
10	8, 9	jca 512	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶))
11	7, 10	impbii 208	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ∈ 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106

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-ex 1783  df-cleq 2730  df-clel 2816

This theorem is referenced by:  inxpxrn  36521