Theorem iuneq12d 3994

Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Hypotheses

Ref	Expression
iuneq1d.1	⊢ (φ → A = B)
iuneq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
iuneq12d	⊢ (φ → ∪x ∈ A C = ∪x ∈ B D)

Distinct variable groups:   x,A   x,B   φ,x

Allowed substitution hints:   C(x)   D(x)

Proof of Theorem iuneq12d

Step	Hyp	Ref	Expression
1		iuneq1d.1	. . 3 ⊢ (φ → A = B)
2	1	iuneq1d 3993	. 2 ⊢ (φ → ∪x ∈ A C = ∪x ∈ B C)
3		iuneq12d.2	. . . 4 ⊢ (φ → C = D)
4	3	adantr 451	. . 3 ⊢ ((φ ∧ x ∈ B) → C = D)
5	4	iuneq2dv 3991	. 2 ⊢ (φ → ∪x ∈ B C = ∪x ∈ B D)
6	2, 5	eqtrd 2385	1 ⊢ (φ → ∪x ∈ A C = ∪x ∈ B D)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iun 3972

This theorem is referenced by: (None)