Theorem 3imtr3 191

Description: A mixed syllogism inference, useful for removing a definition from both sides of an implication.

Hypotheses

Ref	Expression
3imtr3.1	⊢ (φ → ψ)
3imtr3.2	⊢ (φ ↔ χ)
3imtr3.3	⊢ (ψ ↔ θ)

Assertion

Ref	Expression
3imtr3	⊢ (χ → θ)

Proof of Theorem 3imtr3

Step	Hyp	Ref	Expression
1		3imtr3.2	. . 3 ⊢ (φ ↔ χ)
2		3imtr3.1	. . 3 ⊢ (φ → ψ)
3	1, 2	sylbir 176	. 2 ⊢ (χ → ψ)
4		3imtr3.3	. 2 ⊢ (ψ ↔ θ)
5	3, 4	sylib 173	1 ⊢ (χ → θ)

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  3imtr3g 425  sbal 997  onminex 2275  tfinds2 2405  funeu2 2686  idssen 3309  xpen 3383  rankss 3531  distrlem3pr 3923  nnwos 4610  climunii 4883  hlimunii 5143

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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