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Theorem sbco2d 1837
 Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
sbco2d.1 (φxφ)
sbco2d.2 (φzφ)
sbco2d.3 (φ → (ψzψ))
Assertion
Ref Expression
sbco2d (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))

Proof of Theorem sbco2d
StepHypRef Expression
1 sbco2d.2 . . . . 5 (φzφ)
2 sbco2d.3 . . . . 5 (φ → (ψzψ))
31, 2hbim1 1459 . . . 4 ((φψ) → z(φψ))
43sbco2h 1835 . . 3 ([y / z][z / x](φψ) ↔ [y / x](φψ))
5 sbco2d.1 . . . . . 6 (φxφ)
65sbrim 1827 . . . . 5 ([z / x](φψ) ↔ (φ → [z / x]ψ))
76sbbii 1645 . . . 4 ([y / z][z / x](φψ) ↔ [y / z](φ → [z / x]ψ))
81sbrim 1827 . . . 4 ([y / z](φ → [z / x]ψ) ↔ (φ → [y / z][z / x]ψ))
97, 8bitri 173 . . 3 ([y / z][z / x](φψ) ↔ (φ → [y / z][z / x]ψ))
105sbrim 1827 . . 3 ([y / x](φψ) ↔ (φ → [y / x]ψ))
114, 9, 103bitr3i 199 . 2 ((φ → [y / z][z / x]ψ) ↔ (φ → [y / x]ψ))
1211pm5.74ri 170 1 (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643 This theorem is referenced by: (None)
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