Theorem axreplem 5166

Description: Lemma for axrep2 5167 and axrep3 5168. (Contributed by BJ, 6-Aug-2022.)

Assertion

Ref	Expression
axreplem	⊢ (𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒)))))

Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣   𝑥,𝑤   𝑦,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem axreplem

Step	Hyp	Ref	Expression
1		elequ2 2127	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2	1	anbi1d 633	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑧 ∈ 𝑥 ∧ 𝜒) ↔ (𝑧 ∈ 𝑦 ∧ 𝜒)))
3	2	exbidv 1929	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒) ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒)))
4	3	bibi2d 346	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒)) ↔ (𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒))))
5	4	albidv 1928	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒)) ↔ ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒))))
6	5	imbi2d 344	. 2 ⊢ (𝑥 = 𝑦 → ((𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒))) ↔ (𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒)))))
7	6	exbidv 1929	1 ⊢ (𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑥 ∧ 𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧 ∈ 𝑦 ∧ 𝜒)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-9 2122

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788

This theorem is referenced by:  axrep2  5167  axrep3  5168