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Theorem 2sb6 2432
 Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2sb6 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb6
StepHypRef Expression
1 sb6 2417 . 2 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑))
2 19.21v 1855 . . . 4 (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
3 impexp 461 . . . . 5 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
43albii 1737 . . . 4 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
5 sb6 2417 . . . . 5 ([𝑤 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑤𝜑))
65imbi2i 325 . . . 4 ((𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
72, 4, 63bitr4ri 292 . . 3 ((𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
87albii 1737 . 2 (∀𝑥(𝑥 = 𝑧 → [𝑤 / 𝑦]𝜑) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
91, 8bitri 263 1 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  sbcom2  2433  2exsb  2457  2eu6  2546
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