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Theorem sbc2ieOLD 3856
Description: Obsolete version of sbc2ie 3855 as of 12-Oct-2024. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
sbc2ieOLD.1 𝐴 ∈ V
sbc2ieOLD.2 𝐵 ∈ V
sbc2ieOLD.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
sbc2ieOLD ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2ieOLD
StepHypRef Expression
1 sbc2ieOLD.1 . 2 𝐴 ∈ V
2 sbc2ieOLD.2 . 2 𝐵 ∈ V
3 nfv 1909 . . 3 𝑥𝜓
4 nfv 1909 . . 3 𝑦𝜓
52nfth 1795 . . 3 𝑥 𝐵 ∈ V
6 sbc2ieOLD.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
73, 4, 5, 6sbc2iegf 3854 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
81, 2, 7mp2an 689 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  [wsbc 3772
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-sbc 3773
This theorem is referenced by: (None)
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