Metamath Proof Explorer


Theorem rexbidar

Description: More general form of rexbida . (Contributed by Andrew Salmon, 25-Jul-2011)

Ref Expression
Hypotheses ralbidar.1 ( 𝜑 → ∀ 𝑥𝐴 𝜑 )
ralbidar.2 ( ( 𝜑𝑥𝐴 ) → ( 𝜓𝜒 ) )
Assertion rexbidar ( 𝜑 → ( ∃ 𝑥𝐴 𝜓 ↔ ∃ 𝑥𝐴 𝜒 ) )

Proof

Step Hyp Ref Expression
1 ralbidar.1 ( 𝜑 → ∀ 𝑥𝐴 𝜑 )
2 ralbidar.2 ( ( 𝜑𝑥𝐴 ) → ( 𝜓𝜒 ) )
3 2 ex ( 𝜑 → ( 𝑥𝐴 → ( 𝜓𝜒 ) ) )
4 3 ralimi ( ∀ 𝑥𝐴 𝜑 → ∀ 𝑥𝐴 ( 𝑥𝐴 → ( 𝜓𝜒 ) ) )
5 1 4 syl ( 𝜑 → ∀ 𝑥𝐴 ( 𝑥𝐴 → ( 𝜓𝜒 ) ) )
6 df-ral ( ∀ 𝑥𝐴 ( 𝑥𝐴 → ( 𝜓𝜒 ) ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ( 𝑥𝐴 → ( 𝜓𝜒 ) ) ) )
7 5 6 sylib ( 𝜑 → ∀ 𝑥 ( 𝑥𝐴 → ( 𝑥𝐴 → ( 𝜓𝜒 ) ) ) )
8 pm2.43 ( ( 𝑥𝐴 → ( 𝑥𝐴 → ( 𝜓𝜒 ) ) ) → ( 𝑥𝐴 → ( 𝜓𝜒 ) ) )
9 8 pm5.32d ( ( 𝑥𝐴 → ( 𝑥𝐴 → ( 𝜓𝜒 ) ) ) → ( ( 𝑥𝐴𝜓 ) ↔ ( 𝑥𝐴𝜒 ) ) )
10 9 alimi ( ∀ 𝑥 ( 𝑥𝐴 → ( 𝑥𝐴 → ( 𝜓𝜒 ) ) ) → ∀ 𝑥 ( ( 𝑥𝐴𝜓 ) ↔ ( 𝑥𝐴𝜒 ) ) )
11 exbi ( ∀ 𝑥 ( ( 𝑥𝐴𝜓 ) ↔ ( 𝑥𝐴𝜒 ) ) → ( ∃ 𝑥 ( 𝑥𝐴𝜓 ) ↔ ∃ 𝑥 ( 𝑥𝐴𝜒 ) ) )
12 7 10 11 3syl ( 𝜑 → ( ∃ 𝑥 ( 𝑥𝐴𝜓 ) ↔ ∃ 𝑥 ( 𝑥𝐴𝜒 ) ) )
13 df-rex ( ∃ 𝑥𝐴 𝜓 ↔ ∃ 𝑥 ( 𝑥𝐴𝜓 ) )
14 df-rex ( ∃ 𝑥𝐴 𝜒 ↔ ∃ 𝑥 ( 𝑥𝐴𝜒 ) )
15 12 13 14 3bitr4g ( 𝜑 → ( ∃ 𝑥𝐴 𝜓 ↔ ∃ 𝑥𝐴 𝜒 ) )