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Theorem 2sb6rfOLD 2455
 Description: Obsolete version of 2sb6rf 2454 as of 13-Apr-2023. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
2sb5rf.1 𝑧𝜑
2sb5rf.2 𝑤𝜑
Assertion
Ref Expression
2sb6rfOLD (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Distinct variable group:   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb6rfOLD
StepHypRef Expression
1 sbequ12r 2217 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
2 sbequ12r 2217 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑𝜑))
31, 2sylan9bb 510 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜑))
43pm5.74i 272 . . 3 (((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
542albii 1802 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
6 2sb5rf.2 . . . . 5 𝑤𝜑
7619.23 2176 . . . 4 (∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
87albii 1801 . . 3 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
9 2sb5rf.1 . . . 4 𝑧𝜑
10919.23 2176 . . 3 (∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
118, 10bitri 276 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
12 2ax6e 2451 . . 3 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
13 pm5.5 363 . . 3 (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → ((∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ 𝜑))
1412, 13ax-mp 5 . 2 ((∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ 𝜑)
155, 11, 143bitrri 299 1 (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761  Ⅎwnf 1765  [wsb 2042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043 This theorem is referenced by: (None)
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