Theorem prtlem5 40011

Description: Lemma for prter1 40030, prter2 40032, prter3 40033 and prtex 40031. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Assertion

Ref	Expression
prtlem5	⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))

Distinct variable groups:   𝑣,𝑢,𝑥,𝑟   𝑢,𝑠,𝑣,𝑥   𝑢,𝐴,𝑣,𝑥

Allowed substitution hints:   𝐴(𝑠, 𝑟)

Proof of Theorem prtlem5

Step	Hyp	Ref	Expression
1		elequ1 2151	. . . 4 ⊢ (𝑢 = 𝑟 → (𝑢 ∈ 𝑥 ↔ 𝑟 ∈ 𝑥))
2		elequ1 2151	. . . 4 ⊢ (𝑣 = 𝑠 → (𝑣 ∈ 𝑥 ↔ 𝑠 ∈ 𝑥))
3	1, 2	bi2anan9r 651	. . 3 ⊢ ((𝑣 = 𝑠 ∧ 𝑢 = 𝑟) → ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥)))
4	3	rexbidv 3186	. 2 ⊢ ((𝑣 = 𝑠 ∧ 𝑢 = 𝑟) → (∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥)))
5	4	2sbievw 2132	1 ⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  [wsb 2098  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-rex 3087

This theorem is used by: (None)