This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ceqsalgALT

Description: Alternate proof of ceqsalg , not using ceqsalt . (Contributed by NM, 29-Oct-2003) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Revised by BJ, 29-Sep-2019) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ceqsalg.1 ⊢ Ⅎ 𝑥 𝜓
ceqsalg.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion ceqsalgALT ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 ceqsalg.1 ⊢ Ⅎ 𝑥 𝜓
2 ceqsalg.2 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3 elisset ⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
4 nfa1 ⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 )
5 2 biimpd ⊢ ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) )
6 5 a2i ⊢ ( ( 𝑥 = 𝐴 → 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) )
7 6 sps ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) )
8 4 1 7 exlimd ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
9 3 8 syl5com ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → 𝜓 ) )
10 2 biimprcd ⊢ ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) )
11 1 10 alrimi ⊢ ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) )
12 9 11 impbid1 ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )