Theorem rexprgf 4656

Description: Convert a restricted existential quantification over a pair to a disjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011.) (Revised by AV, 2-Apr-2023.)

Hypotheses

Ref	Expression
ralprgf.1	⊢ Ⅎ𝑥𝜓
ralprgf.2	⊢ Ⅎ𝑥𝜒
ralprgf.a	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ralprgf.b	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem rexprgf

Step	Hyp	Ref	Expression
1		df-pr 4587	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2	1	rexeqi 3319	. . 3 ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3		rexun 4142	. . 3 ⊢ (∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑))
4	2, 3	bitri 278	. 2 ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑))
5		ralprgf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
6		ralprgf.a	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	rexsngf 4633	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
8	7	orbi1d 930	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ ∃𝑥 ∈ {𝐵}𝜑)))
9		ralprgf.2	. . . . 5 ⊢ Ⅎ𝑥𝜒
10		ralprgf.b	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
11	9, 10	rexsngf 4633	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (∃𝑥 ∈ {𝐵}𝜑 ↔ 𝜒))
12	11	orbi2d 929	. . 3 ⊢ (𝐵 ∈ 𝑊 → ((𝜓 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ 𝜒)))
13	8, 12	sylan9bb 519	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ 𝜒)))
14	4, 13	bitrid 286	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  ∃wrex 3087   ∪ cun 3897  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3088  df-v 3453  df-sbc 3740  df-un 3904  df-sn 4585  df-pr 4587

This theorem is used by:  reuprg0  4663