Theorem rextpg 3778

Description: Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypotheses

Ref	Expression
ralprg.1	⊢ (x = A → (φ ↔ ψ))
ralprg.2	⊢ (x = B → (φ ↔ χ))
raltpg.3	⊢ (x = C → (φ ↔ θ))

Assertion

Ref	Expression
rextpg	⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (∃x ∈ {A, B, C}φ ↔ (ψ ∨ χ ∨ θ)))

Distinct variable groups:   x,A   x,B   x,C   ψ,x   χ,x   θ,x

Allowed substitution hints:   φ(x)   V(x)   W(x)   X(x)

Proof of Theorem rextpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
2		ralprg.2	. . . . . 6 ⊢ (x = B → (φ ↔ χ))
3	1, 2	rexprg 3776	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ W) → (∃x ∈ {A, B}φ ↔ (ψ ∨ χ)))
4	3	orbi1d 683	. . . 4 ⊢ ((A ∈ V ∧ B ∈ W) → ((∃x ∈ {A, B}φ ∨ ∃x ∈ {C}φ) ↔ ((ψ ∨ χ) ∨ ∃x ∈ {C}φ)))
5		raltpg.3	. . . . . 6 ⊢ (x = C → (φ ↔ θ))
6	5	rexsng 3766	. . . . 5 ⊢ (C ∈ X → (∃x ∈ {C}φ ↔ θ))
7	6	orbi2d 682	. . . 4 ⊢ (C ∈ X → (((ψ ∨ χ) ∨ ∃x ∈ {C}φ) ↔ ((ψ ∨ χ) ∨ θ)))
8	4, 7	sylan9bb 680	. . 3 ⊢ (((A ∈ V ∧ B ∈ W) ∧ C ∈ X) → ((∃x ∈ {A, B}φ ∨ ∃x ∈ {C}φ) ↔ ((ψ ∨ χ) ∨ θ)))
9	8	3impa 1146	. 2 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → ((∃x ∈ {A, B}φ ∨ ∃x ∈ {C}φ) ↔ ((ψ ∨ χ) ∨ θ)))
10		df-tp 3743	. . . 4 ⊢ {A, B, C} = ({A, B} ∪ {C})
11	10	rexeqi 2812	. . 3 ⊢ (∃x ∈ {A, B, C}φ ↔ ∃x ∈ ({A, B} ∪ {C})φ)
12		rexun 3443	. . 3 ⊢ (∃x ∈ ({A, B} ∪ {C})φ ↔ (∃x ∈ {A, B}φ ∨ ∃x ∈ {C}φ))
13	11, 12	bitri 240	. 2 ⊢ (∃x ∈ {A, B, C}φ ↔ (∃x ∈ {A, B}φ ∨ ∃x ∈ {C}φ))
14		df-3or 935	. 2 ⊢ ((ψ ∨ χ ∨ θ) ↔ ((ψ ∨ χ) ∨ θ))
15	9, 13, 14	3bitr4g 279	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (∃x ∈ {A, B, C}φ ↔ (ψ ∨ χ ∨ θ)))

Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∨ w3o 933   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207  {csn 3737  {cpr 3738  {ctp 3739

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-3or 935  df-3an 936  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 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by:  rextp  3782