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Theorem sbco2d 2087
 Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
sbco2d.1 xφ
sbco2d.2 zφ
sbco2d.3 (φ → Ⅎzψ)
Assertion
Ref Expression
sbco2d (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))

Proof of Theorem sbco2d
StepHypRef Expression
1 sbco2d.2 . . . . 5 zφ
2 sbco2d.3 . . . . 5 (φ → Ⅎzψ)
31, 2nfim1 1811 . . . 4 z(φψ)
43sbco2 2086 . . 3 ([y / z][z / x](φψ) ↔ [y / x](φψ))
5 sbco2d.1 . . . . . 6 xφ
65sbrim 2067 . . . . 5 ([z / x](φψ) ↔ (φ → [z / x]ψ))
76sbbii 1653 . . . 4 ([y / z][z / x](φψ) ↔ [y / z](φ → [z / x]ψ))
81sbrim 2067 . . . 4 ([y / z](φ → [z / x]ψ) ↔ (φ → [y / z][z / x]ψ))
97, 8bitri 240 . . 3 ([y / z][z / x](φψ) ↔ (φ → [y / z][z / x]ψ))
105sbrim 2067 . . 3 ([y / x](φψ) ↔ (φ → [y / x]ψ))
114, 9, 103bitr3i 266 . 2 ((φ → [y / z][z / x]ψ) ↔ (φ → [y / x]ψ))
1211pm5.74ri 237 1 (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  [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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbco3  2088
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