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Theorem cbval2vOLD 2357
Description: Obsolete version of cbval2v 2356 as of 14-Jan-2024. (Contributed by BJ, 16-Jan-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbval2v.1 𝑧𝜑
cbval2v.2 𝑤𝜑
cbval2v.3 𝑥𝜓
cbval2v.4 𝑦𝜓
cbval2v.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbval2vOLD (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbval2vOLD
StepHypRef Expression
1 cbval2v.1 . . 3 𝑧𝜑
21nfal 2335 . 2 𝑧𝑦𝜑
3 cbval2v.3 . . 3 𝑥𝜓
43nfal 2335 . 2 𝑥𝑤𝜓
5 nfv 1908 . . . . . 6 𝑤 𝑥 = 𝑧
6 cbval2v.2 . . . . . 6 𝑤𝜑
75, 6nfim 1890 . . . . 5 𝑤(𝑥 = 𝑧𝜑)
8 nfv 1908 . . . . . 6 𝑦 𝑥 = 𝑧
9 cbval2v.4 . . . . . 6 𝑦𝜓
108, 9nfim 1890 . . . . 5 𝑦(𝑥 = 𝑧𝜓)
11 cbval2v.5 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1211expcom 416 . . . . . 6 (𝑦 = 𝑤 → (𝑥 = 𝑧 → (𝜑𝜓)))
1312pm5.74d 275 . . . . 5 (𝑦 = 𝑤 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧𝜓)))
147, 10, 13cbvalv1 2354 . . . 4 (∀𝑦(𝑥 = 𝑧𝜑) ↔ ∀𝑤(𝑥 = 𝑧𝜓))
15 19.21v 1933 . . . 4 (∀𝑦(𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦𝜑))
16 19.21v 1933 . . . 4 (∀𝑤(𝑥 = 𝑧𝜓) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
1714, 15, 163bitr3i 303 . . 3 ((𝑥 = 𝑧 → ∀𝑦𝜑) ↔ (𝑥 = 𝑧 → ∀𝑤𝜓))
1817pm5.74ri 274 . 2 (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
192, 4, 18cbvalv1 2354 1 (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528  wnf 1777
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-10 2138  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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