Theorem 3r19.43 3137

Description: Restricted quantifier version of 19.43 1915 for a triple disjunction . (Contributed by AV, 2-Nov-2025.)

Assertion

Ref	Expression
3r19.43	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∃𝑥 ∈ 𝐴 𝜒))

Proof of Theorem 3r19.43

Step	Hyp	Ref	Expression
1		df-3or 1104	. . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒))
2	1	rexbii 3115	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ∃𝑥 ∈ 𝐴 ((𝜑 ∨ 𝜓) ∨ 𝜒))
3		r19.43 3136	. 2 ⊢ (∃𝑥 ∈ 𝐴 ((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∨ ∃𝑥 ∈ 𝐴 𝜒))
4		r19.43 3136	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
5	4	orbi1i 927	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∨ ∃𝑥 ∈ 𝐴 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ∨ ∃𝑥 ∈ 𝐴 𝜒))
6		df-3or 1104	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∃𝑥 ∈ 𝐴 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ∨ ∃𝑥 ∈ 𝐴 𝜒))
7	5, 6	bitr4i 281	. 2 ⊢ ((∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∨ ∃𝑥 ∈ 𝐴 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∃𝑥 ∈ 𝐴 𝜒))
8	2, 3, 7	3bitri 300	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓 ∨ ∃𝑥 ∈ 𝐴 𝜒))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   ∨ w3o 1102  ∃wrex 3092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-ex 1813  df-ral 3083  df-rex 3093

This theorem is used by:  gpgprismgriedgdmss  48858