Theorem 19.9d2r 32890

Description: A deduction version of one direction of 19.9 2244 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypotheses

Ref	Expression
19.9d2r.1	⊢ (𝜑 → Ⅎ𝑥𝜓)
19.9d2r.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
19.9d2r.3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Assertion

Ref	Expression
19.9d2r	⊢ (𝜑 → 𝜓)

Distinct variable group:   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥, 𝑦)   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)

Proof of Theorem 19.9d2r

Step	Hyp	Ref	Expression
1		nfv 1947	. 2 ⊢ Ⅎ𝑦𝜑
2		19.9d2r.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		19.9d2r.2	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		19.9d2r.3	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
5	1, 2, 3, 4	19.9d2rf 32889	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1816  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-rex 3092

This theorem is used by:  xrofsup  33184