Theorem rexraleqim 3579

Description: Statement following from existence and generalization with equality. (Contributed by AV, 9-Feb-2019.)

Hypotheses

Ref	Expression
rexraleqim.1	⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜑))
rexraleqim.2	⊢ (𝑧 = 𝑌 → (𝜑 ↔ 𝜃))

Assertion

Ref	Expression
rexraleqim	⊢ ((∃𝑧 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → 𝜃)

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝑌,𝑧   𝜑,𝑥   𝜓,𝑧   𝜃,𝑧

Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥)   𝜃(𝑥)

Proof of Theorem rexraleqim

Step	Hyp	Ref	Expression
1		rexraleqim.1	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ 𝜑))
2		eqeq1 2799	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑌 ↔ 𝑧 = 𝑌))
3	1, 2	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝑌) ↔ (𝜑 → 𝑧 = 𝑌)))
4	3	rspcva 3557	. . . . 5 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → (𝜑 → 𝑧 = 𝑌))
5		rexraleqim.2	. . . . . 6 ⊢ (𝑧 = 𝑌 → (𝜑 ↔ 𝜃))
6	5	biimpd 230	. . . . 5 ⊢ (𝑧 = 𝑌 → (𝜑 → 𝜃))
7	4, 6	syli 39	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → (𝜑 → 𝜃))
8	7	impancom 452	. . 3 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝜑) → (∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌) → 𝜃))
9	8	rexlimiva 3244	. 2 ⊢ (∃𝑧 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌) → 𝜃))
10	9	imp 407	1 ⊢ ((∃𝑧 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑌)) → 𝜃)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439

This theorem is referenced by:  cramerlem3  20982