Theorem bibi12i 462

Description: The equivalence of two equivalences.

Hypotheses

Ref	Expression
bibi12.1	⊢ (φ ↔ ψ)
bibi12.2	⊢ (χ ↔ θ)

Assertion

Ref	Expression
bibi12i	⊢ ((φ ↔ χ) ↔ (ψ ↔ θ))

Proof of Theorem bibi12i

Step	Hyp	Ref	Expression
1		bibi12.2	. . 3 ⊢ (χ ↔ θ)
2	1	bibi2i 460	. 2 ⊢ ((φ ↔ χ) ↔ (φ ↔ θ))
3		bibi12.1	. . 3 ⊢ (φ ↔ ψ)
4	3	bibi1i 461	. 2 ⊢ ((φ ↔ θ) ↔ (ψ ↔ θ))
5	2, 4	bitr 151	1 ⊢ ((φ ↔ χ) ↔ (ψ ↔ θ))

Syntax hints:   ↔ wb 127

This theorem is referenced by:  pm5.32 488  biluk 512  cleq2ab 1179  dmcosseq 2572  fv2 2828  zfcndrep 3760

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  df-an 198

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