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Theorem ralprgOLD 4694
Description: Obsolete version of ralprg 4693 as of 30-Sep-2024. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) (Proof shortened by AV, 8-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ralprg.1 (𝑥 = 𝐴 → (𝜑𝜓))
ralprg.2 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
ralprgOLD ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ralprgOLD
StepHypRef Expression
1 nfv 1909 . 2 𝑥𝜓
2 nfv 1909 . 2 𝑥𝜒
3 ralprg.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
4 ralprg.2 . 2 (𝑥 = 𝐵 → (𝜑𝜒))
51, 2, 3, 4ralprgf 4691 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  {cpr 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-v 3470  df-sbc 3773  df-un 3948  df-sn 4624  df-pr 4626
This theorem is referenced by: (None)
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