Theorem 19.9d2rf 32220

Description: A deduction version of one direction of 19.9 2190 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)

Hypotheses

Ref	Expression
19.9d2rf.0	⊢ Ⅎ𝑦𝜑
19.9d2rf.1	⊢ (𝜑 → Ⅎ𝑥𝜓)
19.9d2rf.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
19.9d2rf.3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)

Assertion

Ref	Expression
19.9d2rf	⊢ (𝜑 → 𝜓)

Proof of Theorem 19.9d2rf

Step	Hyp	Ref	Expression
1		19.9d2rf.3	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
2		rexex 3070	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥∃𝑦 ∈ 𝐵 𝜓)
3		rexex 3070	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜓 → ∃𝑦𝜓)
4	3	eximi 1829	. . . 4 ⊢ (∃𝑥∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥∃𝑦𝜓)
5	1, 2, 4	3syl 18	. . 3 ⊢ (𝜑 → ∃𝑥∃𝑦𝜓)
6		19.9d2rf.0	. . . . 5 ⊢ Ⅎ𝑦𝜑
7		19.9d2rf.1	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
8	6, 7	nfexd 2316	. . . 4 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)
9	8	19.9d 2188	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓))
10	5, 9	mpd 15	. 2 ⊢ (𝜑 → ∃𝑦𝜓)
11		19.9d2rf.2	. . 3 ⊢ (𝜑 → Ⅎ𝑦𝜓)
12	11	19.9d 2188	. 2 ⊢ (𝜑 → (∃𝑦𝜓 → 𝜓))
13	10, 12	mpd 15	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4  ∃wex 1773  Ⅎwnf 1777  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-rex 3065

This theorem is referenced by:  19.9d2r  32221  xrofsup  32487