Theorem rexprgf 4694

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 4628	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2	1	rexeqi 3324	. . 3 ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3		rexun 4195	. . 3 ⊢ (∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑))
4	2, 3	bitri 275	. 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 4671	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
8	7	orbi1d 916	. . 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 4671	. . . 4 ⊢ (𝐵 ∈ 𝑊 → (∃𝑥 ∈ {𝐵}𝜑 ↔ 𝜒))
12	11	orbi2d 915	. . 3 ⊢ (𝐵 ∈ 𝑊 → ((𝜓 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ 𝜒)))
13	8, 12	sylan9bb 509	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ 𝜒)))
14	4, 13	bitrid 283	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539  Ⅎwnf 1782   ∈ wcel 2107  ∃wrex 3069   ∪ cun 3948  {csn 4625  {cpr 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rex 3070  df-v 3481  df-sbc 3788  df-un 3955  df-sn 4626  df-pr 4628

This theorem is referenced by:  reuprg0  4701