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Theorem cbvrexdva2OLD 3457
Description: Obsolete version of cbvrexdva 3459 as of 12-Aug-2023. (Contributed by David Moews, 1-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvraldva2.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
cbvraldva2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvrexdva2OLD (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvrexdva2OLD
StepHypRef Expression
1 simpr 487 . . . . . . . . . . . 12 ((𝜑𝑥 = 𝑦) → 𝑥 = 𝑦)
2 cbvraldva2.2 . . . . . . . . . . . 12 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
31, 2eleq12d 2905 . . . . . . . . . . 11 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
4 cbvraldva2.1 . . . . . . . . . . 11 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
53, 4anbi12d 632 . . . . . . . . . 10 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝜓) ↔ (𝑦𝐵𝜒)))
65notbid 320 . . . . . . . . 9 ((𝜑𝑥 = 𝑦) → (¬ (𝑥𝐴𝜓) ↔ ¬ (𝑦𝐵𝜒)))
76expcom 416 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑 → (¬ (𝑥𝐴𝜓) ↔ ¬ (𝑦𝐵𝜒))))
87pm5.74d 275 . . . . . . 7 (𝑥 = 𝑦 → ((𝜑 → ¬ (𝑥𝐴𝜓)) ↔ (𝜑 → ¬ (𝑦𝐵𝜒))))
98cbvalvw 2036 . . . . . 6 (∀𝑥(𝜑 → ¬ (𝑥𝐴𝜓)) ↔ ∀𝑦(𝜑 → ¬ (𝑦𝐵𝜒)))
10 19.21v 1933 . . . . . 6 (∀𝑥(𝜑 → ¬ (𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑥 ¬ (𝑥𝐴𝜓)))
11 19.21v 1933 . . . . . 6 (∀𝑦(𝜑 → ¬ (𝑦𝐵𝜒)) ↔ (𝜑 → ∀𝑦 ¬ (𝑦𝐵𝜒)))
129, 10, 113bitr3i 303 . . . . 5 ((𝜑 → ∀𝑥 ¬ (𝑥𝐴𝜓)) ↔ (𝜑 → ∀𝑦 ¬ (𝑦𝐵𝜒)))
1312pm5.74ri 274 . . . 4 (𝜑 → (∀𝑥 ¬ (𝑥𝐴𝜓) ↔ ∀𝑦 ¬ (𝑦𝐵𝜒)))
14 alnex 1775 . . . 4 (∀𝑥 ¬ (𝑥𝐴𝜓) ↔ ¬ ∃𝑥(𝑥𝐴𝜓))
15 alnex 1775 . . . 4 (∀𝑦 ¬ (𝑦𝐵𝜒) ↔ ¬ ∃𝑦(𝑦𝐵𝜒))
1613, 14, 153bitr3g 315 . . 3 (𝜑 → (¬ ∃𝑥(𝑥𝐴𝜓) ↔ ¬ ∃𝑦(𝑦𝐵𝜒)))
1716con4bid 319 . 2 (𝜑 → (∃𝑥(𝑥𝐴𝜓) ↔ ∃𝑦(𝑦𝐵𝜒)))
18 df-rex 3142 . 2 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
19 df-rex 3142 . 2 (∃𝑦𝐵 𝜒 ↔ ∃𝑦(𝑦𝐵𝜒))
2017, 18, 193bitr4g 316 1 (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1528   = wceq 1530  wex 1773  wcel 2107  wrex 3137
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-cleq 2812  df-clel 2891  df-rex 3142
This theorem is referenced by: (None)
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