Theorem phphashd

Description: Corollary of the Pigeonhole Principle using equality. Equivalent of phpeqd expressed using the hash function. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	phphashd.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	phphashd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
	phphashd.3	⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) )
Assertion	phphashd	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		phphashd.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		phphashd.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3		phphashd.3	⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) )
4	1 2	ssfid	⊢ ( 𝜑 → 𝐵 ∈ Fin )
5		hashen	⊢ ( ( 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ) → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )
6	1 4 5	syl2anc	⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) = ( ♯ ‘ 𝐵 ) ↔ 𝐴 ≈ 𝐵 ) )
7	3 6	mpbid	⊢ ( 𝜑 → 𝐴 ≈ 𝐵 )
8	1 2 7	phpeqd	⊢ ( 𝜑 → 𝐴 = 𝐵 )