Theorem ixpeq2 8905

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 8904	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)
2		ss2ixp 8904	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵)
3	1, 2	anim12i 614	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶 ∧ X𝑥 ∈ 𝐴 𝐶 ⊆ X𝑥 ∈ 𝐴 𝐵))
4		eqss 3998	. . . 4 ⊢ (𝐵 = 𝐶 ↔ (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
5	4	ralbii 3094	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 ↔ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵))
6		r19.26 3112	. . 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 3998	. 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 397   = wceq 1542  ∀wral 3062   ⊆ wss 3949  Xcixp 8891

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-ral 3063  df-v 3477  df-in 3956  df-ss 3966  df-ixp 8892

This theorem is referenced by:  ixpeq2dva  8906  ixpint  8919  prdsbas3  17427  pwsbas  17433  ptbasfi  23085  ptunimpt  23099  pttopon  23100  ptcld  23117  ptrescn  23143  ptuncnv  23311  ptunhmeo  23312  ptrest  36535  prdstotbnd  36710  ixpeq2d  43803  hoidmv1le  45358