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Theorem 2mos 2283
 Description: Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
Hypothesis
Ref Expression
2mos.1 ((x = z y = w) → (φψ))
Assertion
Ref Expression
2mos (zwxy(φ → (x = z y = w)) ↔ xyzw((φ ψ) → (x = z y = w)))
Distinct variable groups:   z,w,φ   x,y,ψ   x,z,w,y
Allowed substitution hints:   φ(x,y)   ψ(z,w)

Proof of Theorem 2mos
StepHypRef Expression
1 2mo 2282 . 2 (zwxy(φ → (x = z y = w)) ↔ xyzw((φ [z / x][w / y]φ) → (x = z y = w)))
2 nfv 1619 . . . . . . 7 xψ
3 nfv 1619 . . . . . . . . . 10 y x = z
43sbrim 2067 . . . . . . . . 9 ([w / y](x = zφ) ↔ (x = z → [w / y]φ))
5 nfv 1619 . . . . . . . . . 10 y(x = zψ)
6 2mos.1 . . . . . . . . . . . 12 ((x = z y = w) → (φψ))
76expcom 424 . . . . . . . . . . 11 (y = w → (x = z → (φψ)))
87pm5.74d 238 . . . . . . . . . 10 (y = w → ((x = zφ) ↔ (x = zψ)))
95, 8sbie 2038 . . . . . . . . 9 ([w / y](x = zφ) ↔ (x = zψ))
104, 9bitr3i 242 . . . . . . . 8 ((x = z → [w / y]φ) ↔ (x = zψ))
1110pm5.74ri 237 . . . . . . 7 (x = z → ([w / y]φψ))
122, 11sbie 2038 . . . . . 6 ([z / x][w / y]φψ)
1312anbi2i 675 . . . . 5 ((φ [z / x][w / y]φ) ↔ (φ ψ))
1413imbi1i 315 . . . 4 (((φ [z / x][w / y]φ) → (x = z y = w)) ↔ ((φ ψ) → (x = z y = w)))
15142albii 1567 . . 3 (zw((φ [z / x][w / y]φ) → (x = z y = w)) ↔ zw((φ ψ) → (x = z y = w)))
16152albii 1567 . 2 (xyzw((φ [z / x][w / y]φ) → (x = z y = w)) ↔ xyzw((φ ψ) → (x = z y = w)))
171, 16bitri 240 1 (zwxy(φ → (x = z y = w)) ↔ xyzw((φ ψ) → (x = z y = w)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by: (None)
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