Theorem 3imtr3d 420

Description: More general version of 3imtr3 191. Useful for converting conditional definitions in a formula.

Hypotheses

Ref	Expression
3imtr3d.1	⊢ (φ → (ψ → χ))
3imtr3d.2	⊢ (φ → (ψ ↔ θ))
3imtr3d.3	⊢ (φ → (χ ↔ τ))

Assertion

Ref	Expression
3imtr3d	⊢ (φ → (θ → τ))

Proof of Theorem 3imtr3d

Step	Hyp	Ref	Expression
1		3imtr3d.2	. 2 ⊢ (φ → (ψ ↔ θ))
2		3imtr3d.1	. . 3 ⊢ (φ → (ψ → χ))
3		3imtr3d.3	. . 3 ⊢ (φ → (χ ↔ τ))
4	2, 3	sylibd 177	. 2 ⊢ (φ → (ψ → τ))
5	1, 4	sylbird 180	1 ⊢ (φ → (θ → τ))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  ddelimdf 909  fornex 2793  sdomel 3653  cardsdomel 3658  ltapr 3945  nnleltp1t 4448  pjnormss 5638

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128

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