Theorem ixpeq2 8847

Description: Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.)

Assertion

Ref	Expression
ixpeq2	⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Proof of Theorem ixpeq2

Step	Hyp	Ref	Expression
1		ss2ixp 8846	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
2		ss2ixp 8846	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵)
3	1, 2	anim12i 613	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶 ∧ X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵))
4		eqss 3947	. . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
5	4	ralbii 3080	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
6		r19.26 3094	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
7	5, 6	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵))
8		eqss 3947	. 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶 ↔ (X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶 ∧ X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵))
9	3, 7, 8	3imtr4i 292	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∀wral 3049   ⊆ wss 3899  Xcixp 8833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-ss 3916  df-ixp 8834

This theorem is referenced by:  ixpeq2dva  8848  ixpint  8861  prdsbas3  17399  pwsbas  17405  ptbasfi  23523  ptunimpt  23537  pttopon  23538  ptcld  23555  ptrescn  23581  ptuncnv  23749  ptunhmeo  23750  ixpeq12i  36344  ptrest  37759  prdstotbnd  37934  ixpeq2d  45255  hoidmv1le  46780