Metamath Proof Explorer


Theorem ceqsralv2

Description: Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020)

Ref Expression
Hypothesis ceqsralv2.1 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
Assertion ceqsralv2 ( ∀ 𝑥𝐵 ( 𝑥 = 𝐴𝜑 ) ↔ ( 𝐴𝐵𝜓 ) )

Proof

Step Hyp Ref Expression
1 ceqsralv2.1 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
2 1 notbid ( 𝑥 = 𝐴 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3 2 ceqsrexv2 ( ∃ 𝑥𝐵 ( 𝑥 = 𝐴 ∧ ¬ 𝜑 ) ↔ ( 𝐴𝐵 ∧ ¬ 𝜓 ) )
4 rexanali ( ∃ 𝑥𝐵 ( 𝑥 = 𝐴 ∧ ¬ 𝜑 ) ↔ ¬ ∀ 𝑥𝐵 ( 𝑥 = 𝐴𝜑 ) )
5 annim ( ( 𝐴𝐵 ∧ ¬ 𝜓 ) ↔ ¬ ( 𝐴𝐵𝜓 ) )
6 3 4 5 3bitr3i ( ¬ ∀ 𝑥𝐵 ( 𝑥 = 𝐴𝜑 ) ↔ ¬ ( 𝐴𝐵𝜓 ) )
7 6 con4bii ( ∀ 𝑥𝐵 ( 𝑥 = 𝐴𝜑 ) ↔ ( 𝐴𝐵𝜓 ) )